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Similar Question 1
<p> Try to solve a triangle with side lengths <code class='latex inline'> 3 \mathrm{~cm}, 8 \mathrm{~cm} </code> , and <code class='latex inline'> 4 \mathrm{~cm} </code> using the cosine law. What do you find? Explain why.</p>
Similar Question 2
<p>In <code class='latex inline'>\triangle PQR, \angle P = 80^o, \angle Q = 48^o</code>, and <code class='latex inline'>r = 20 cm</code>. Solve <code class='latex inline'>\triangle PQR</code>.</p>
Similar Question 3
<p>Solve each of the following triangles. Round each answer to the nearest whole number, if necessary.</p><img src="/qimages/66313" />
Similar Questions
Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
Now You Try
<p>Determine whether the primary trigonometric ratios, the sine law, or the cosine law should be used first to solve each triangle.</p><img src="/qimages/23047" />
<p>Find the missing side length in each triangle, to the nearest unit.</p><p><strong>a)</strong></p><img src="/qimages/2517" /><p><strong>b)</strong></p><img src="/qimages/2518" />
<p>One of the tallest totem poles in the world is located in Surrey, British Columbia. When the angle of elevation of the Sun is <code class='latex inline'>62^{\circ}</code>, the totem pole casts a shadow of 30 m.</p><p><strong>a)</strong> Suppose the totem stood vertical, how tall would it be, to the nearest tenth of a metre?</p><p><strong>b)</strong> Suppose it was not quite vertical so that it made an angle of <code class='latex inline'>89</code><code class='latex inline'>^\circ</code> with the ground. In this case, would your answer for its height be taller or shorter than your answer in part a)? Justify without calculations.</p><p>Calculate the difference between these two heights.</p>
<p>Solve the triangle. Round answers to the nearest tenth of a degree.</p><p>In acute <code class='latex inline'>\triangle TUV, t = 10.3 m</code>, <code class='latex inline'>u = 11.4 m</code>, and <code class='latex inline'>v = 12.5 m</code></p>
<p>Solve each triangle. Round answers to the nearest unit, if necessary.</p><p><strong>a)</strong></p><img src="/qimages/2522" /><p><strong>b)</strong></p><img src="/qimages/2523" /><p><strong>c)</strong></p><img src="/qimages/2524" />
<p>A small commercial plane and a jet airliner are 7.5 km from each other, at the same altitude. From an observation tower the two aircraft are separated by an angle of 68<code class='latex inline'>^\circ</code>. If the jet airliner is 5.2 km from the observation tower, how far is the commercial plane from the observation tower, how far is the commercial plane from the observation tower, to the nearest tenth of a kilometre?</p><img src="/qimages/2512" />
<p>Solve. Express answers as integers or as decimals, to the nearest tenth.</p><p><code class='latex inline'>\displaystyle 5 x^{2}=8 x </code></p>
<p>Sketch each triangle and label it with the given information. Then, use the given information to solve the triangle. Round your answers to the nearest tenth of a degree.</p><p>In acute <code class='latex inline'>\triangle DJP</code>, <code class='latex inline'>d = 21.3 mm</code>, <code class='latex inline'>j = 24.6 mm</code>, and <code class='latex inline'>p = 23.7 mm</code>.</p>
<p>Find the length <code class='latex inline'>x</code>, to the nearest tenth of a metre.</p><img src="/qimages/9441" />
<p>Sketch each triangle and label the given information. Then, solve the triangle. Round side lengths to the nearest tenth of a unit and angles to the nearest degree.</p><p>In <code class='latex inline'>\triangle BXR</code>, <code class='latex inline'>\angle X = 54^{\circ}</code>, <code class='latex inline'>b = 17 m</code>, and <code class='latex inline'>r= 12 m</code>.</p>
<p>Solve each triangle. Round answers to the nearest unit, if necessary.</p><img src="/qimages/22902" />
<p>Solve the triangle. Round answers to the nearest tenth of a degree.</p><p>In acute <code class='latex inline'>\triangle SBZ</code>, <code class='latex inline'>s = 19 m, b =21 m</code>, and <code class='latex inline'>z = 13 m</code>.</p>
<p>Find the indicated quantity, to the nearest tenth of a unit.</p><p>In <code class='latex inline'>\displaystyle \triangle \mathrm{KLM}, k=36.5 \mathrm{~m}, m=51.4 \mathrm{~m} </code>, and <code class='latex inline'>\displaystyle \angle \mathrm{L}=72.1^{\circ} . </code> Find <code class='latex inline'>\displaystyle l . </code></p>
<p>Ron and Ben are two koala bears frolicking in a meadow. Suddenly, a tasty clump of eucalyptus falls to the ground, catching their attention. Ben glances at Ron, who appears to be <code class='latex inline'>15</code> m away, then over to the eucalyptus, which appears to be <code class='latex inline'>18</code> m away. From Ben&#39;s point of view, Ron and the eucalyptus are separated by an angle of <code class='latex inline'>45</code><code class='latex inline'>^\circ</code>. Rocco’s top running speed is 1.0 m/s, but Ben can run one and a half times as fast. Can Ben beat Ron to the eucalyptus? State any assumptions you make.</p>
<p>Find the length of the indicated side, to the nearest metre.</p><img src="/qimages/66286" />
<img src="/qimages/10082" /><p>In <code class='latex inline'>\displaystyle \triangle A B C, \angle A=58^{\circ}, b=10.0 \mathrm{~cm} </code>, and <code class='latex inline'>\displaystyle c=14.0 \mathrm{~cm} </code>. Solve <code class='latex inline'>\displaystyle \triangle A B C </code></p><p>A. <code class='latex inline'>\displaystyle \angle B=90^{\circ}, \angle C=32^{\circ}, a=9.8 \mathrm{~cm} </code></p><p>B. <code class='latex inline'>\displaystyle \angle B=86^{\circ}, \angle C=36^{\circ}, a=17.2 \mathrm{~cm} </code></p><p>C. <code class='latex inline'>\displaystyle \angle B=44^{\circ}, \angle C=78^{\circ}, a=12.2 \mathrm{~cm} </code></p><p>D. <code class='latex inline'>\displaystyle \angle B=78^{\circ}, \angle C=44^{\circ}, a=12.2 \mathrm{~cm} </code></p>
<p>Find the measure of the indicated angle in each triangle, to the nearest degree.</p><img src="/qimages/22899" />
<p>An airplane is flying from Montreal to Vancouver. The wind is blowing from the west at 60 km/h. The airplane flies at an airspeed of 750 km/h and must stay on a heading of65° west of north.</p> <ul> <li>What heading should the pilot take to compensate for the wind?</li> </ul>
<img src="/qimages/66326" />
<p>Find the measure of <code class='latex inline'>\angle P</code>, to the nearest degree.</p><img src="/qimages/9472" />
<p>Adam is standing on the level ground directly between two buildings. The top of the first building is 23 m away from Adam, at an angle of elevation of <code class='latex inline'>46^{\circ}</code>, while the top of the second building is 27 m away from Adam at an angle of elevation of <code class='latex inline'>52^{\circ}</code>.</p><p>a) Draw a diagram and label the known information.</p><p>b) How far apart are the tops of the two buildings, to the nearest metre?</p><p>c) How far is Adam from the bottom of each building, to the nearest metre?</p><p>d) What is the height of each building, to the nearest metre?</p>
<p>Find the length of the indicated side in each triangle, to the nearest unit.</p><img src="/qimages/22908" />
<p>Find the measure of the indicated angle in each triangle, to the nearest degree.</p><img src="/qimages/22900" />
<p>Solve the triangle. Round answers to the nearest unit, if necessary.</p><img src="/qimages/9473" />
<p> Use the sine law to explain why the two equal sides of an isosceles triangle must be opposite the two equal angles.</p>
<p>Find the length of the indicated side in each triangle, to the nearest unit.</p><img src="/qimages/22893" />
<p>Draw a diagram and label the given information. Then, find the measure of the indicated side in each triangle, to the nearest degree.</p><p>In acute <code class='latex inline'>\triangle MNO</code>, <code class='latex inline'>\angle M = 68^{\circ}</code>, <code class='latex inline'>m = 18.2 cm</code>, and <code class='latex inline'>n = 15.3 cm</code>. Find <code class='latex inline'>\angle N</code>.</p>
<p> For each of the following acute triangles, find the measure of the indicated angle, to the nearest degree.</p><img src="/qimages/66293" />
<p>Lan’s garden is in the shape of an acute triangle. Two of the vertices have angles of 55<code class='latex inline'>^\circ</code> and <code class='latex inline'>58</code><code class='latex inline'>^\circ</code>. The side joining these vertices is 6.2 m in length. What is the area of this garden, to the nearest tenth of a square metre?</p><p>You may use Heron&#39;s formula:</p><p><code class='latex inline'> \displaystyle A = \sqrt{s(s -a )(s -b)(s- c)} </code> where a, b, and ac are the side lengths, and </p><p><code class='latex inline'> \displaystyle s = \frac{1}{2}(a + b+ c) </code></p>
<p>Cheryl is trying to hit her golf ball between two trees. She estimates the distances shown. Within what angle must Cheryl make her shot, in order to pass between the trees? Round to the nearest tenth of a degree.</p><img src="/qimages/9450" />
<p>Solve each triangle. Round your answers to the nearest tenth of a degree.</p><img src="/qimages/22923" />
<p> Solve each of the following triangles. Round each answer to the nearest whole number, if necessary.</p><img src="/qimages/66314" />
<p>Jane claims that she can draw an acute triangle using the following information: <code class='latex inline'>a = 6 cm, b = 8 cm, c = 10 cm</code>, <code class='latex inline'>\angle A = 30^o</code>, and <code class='latex inline'>\angle B = 60^o</code>. Is she correct? Explain.</p>
<p>Two wires are supporting a tent pole, as shown.</p><img src="/qimages/9449" /><p>a) How far apart are the wires fixed in the ground, to the nearest tenth of a metre?</p><p>b) Find the angle each wire makes with the ground, to the nearest degree. State any assumptions you make.</p>
<p>Draw a diagram and label the given information. Then, solve each triangle. Round answers to the nearest unit, if necessary.</p><p>In <code class='latex inline'>\triangle PSV</code>, <code class='latex inline'>\angle S = 72^{\circ}</code>, <code class='latex inline'>\angle V = 25^{\circ}</code>, and <code class='latex inline'>p = 18 cm</code>.</p>
<p>Which of the following are not correct for acute triangle DEF?</p><p>a) <code class='latex inline'>\displaystyle \frac{d}{\sin D} = \frac{f}{\sin F} </code></p><p>b) <code class='latex inline'>\displaystyle \frac{\sin E}{e} = \frac{\sin D}{d} </code></p>
<p>Find the perimeter of isosceles <code class='latex inline'> \triangle \mathrm{RST} </code> , to the nearest centimetre.</p><img src="/qimages/66325" />
<p>Solve the triangle.</p> <ul> <li>In <code class='latex inline'>\triangle DEF,d=5.0cm, e=6.5cm</code>, and <code class='latex inline'>\angle F</code>=65°.</li> </ul>
<p> Determine whether the sine law or the cosine law should be used first to solve each triangle. Then, solve the triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree.</p><img src="/qimages/66377" />
<p>Find <code class='latex inline'>r</code>.</p><img src="/qimages/9446" />
<p>Draw a diagram and label the given information. Then, find the measure of the indicated side in each triangle, to the nearest tenth of a unit.</p><p>In acute <code class='latex inline'>\triangle DEF</code>, <code class='latex inline'>\angle D = 64^{\circ}</code>, <code class='latex inline'>\angle E = 78^{\circ}</code>, and <code class='latex inline'>d = 15 cm</code>. Find side <code class='latex inline'>e</code>.</p>
<p>Two drivers leave home at the same time and travel on straight roads that diverge by 70°. One driver travels at an average speed of 83.0 km/h. The other driver travels at an average speed of 95.0 km/h. How far apart will the two drivers be after 45 min?</p>
<p>Sketch each triangle and use the given information to find the missing side length, to the nearest tenth of a unit.</p><p>In acute <code class='latex inline'>\triangle MNO</code>, <code class='latex inline'>m = 4.8 cm</code>, <code class='latex inline'>o = 5.9 cm</code>, and <code class='latex inline'>\angle N = 63^{\circ}</code>.</p>
<p>Solve each triangle. Round side lengths to the nearest tenth of a unit and angles to the nearest degree.</p><img src="/qimages/22914" />
<p>Find the length of the tunnel, to the nearest metre.</p><img src="/qimages/9473" />
<p>In acute <code class='latex inline'>\triangle NRC</code>, <code class='latex inline'>\angle N = 47^o</code>, <code class='latex inline'>\angle R =</code> 80°, and <code class='latex inline'>r= 27mm</code>.</p>
<p>Ally is standing in the playing field at the back of his school. From where he is standing, the angle of elevation of the bottom of a flagpole on the roof of the school is <code class='latex inline'>35^{\circ}</code>, while the angle of elevation of the top of the flagpole is <code class='latex inline'>40^{\circ}</code>. The distance from Ally to the bottom of the flagpole is 37 m, and the distance from Ally to the top of the flagpole is 40 m.</p><p>a) Draw a diagram and label the known information.</p><p>b) What is the height of the flagpole, to the nearest tenth of a metre?</p><p>c) How far along the level ground is Ally from the school, to the nearest tenth of a metre?</p>
<p>Determine the indicated side length or angle measure in each triangle.</p><img src="/qimages/10069" />
<p>Ling is designing a garden for her backyard. It will consist of two congruent triangular flower beds on either side of a path, as shown.</p><img src="/qimages/2538" /><p><strong>a)</strong> Find the interior angles of the flower beds, to the nearest degree.</p><p><strong>b)</strong> Find the total area of the flower beds, to the nearest square metre.</p>
<p>In <code class='latex inline'>\triangle ABC</code>, <code class='latex inline'>\angle B = 38.2^o</code>, <code class='latex inline'>\angle C = 65.6^o</code>, and <code class='latex inline'>b = 54 cm</code>. Find <code class='latex inline'>c</code>, to the nearest tenth of a centimetre.</p>
<p>Solve each triangle. Round answers to the nearest unit, if necessary.</p><img src="/qimages/22901" />
<p>In <code class='latex inline'>\triangle ABC</code>, AB = 31°, 5 = 22 cm, and t = 12 cm. Determine <code class='latex inline'>\angle C</code>.</p>
<p>Find the indicated angle in each triangle, to the nearest degree.</p><img src="/qimages/22919" />
<p>Find the measure of the indicated angle in each triangle, to the nearest degree.</p><img src="/qimages/22896" />
<p>From a rock ledge, the angle of elevation to the top of a tree is 22°. The angle of depression to the base of the tree is 12°.</p><img src="/qimages/9463" /><p>a) Find the height of the rock ledge, to the nearest metre.</p><p>b) Find the height of the tree, to the nearest metre.</p>
<p>Find the measure of <code class='latex inline'>\angle P</code>, to the nearest degree.</p><img src="/qimages/5710" />
<p>Find the length of t, to the nearest metre.</p><img src="/qimages/9443" />
<p>Determine whether the primary trigonometric ratios, the sine law, or the cosine law should be used first to solve each triangle.</p><img src="/qimages/23048" />
<p>Determine whether the primary trigonometric ratios, the sine law, or the cosine law should be used first to solve each triangle.</p><img src="/qimages/23049" />
<p>Find the indicated quantity, to the nearest tenth.</p><p>In acute <code class='latex inline'>\displaystyle \triangle \mathrm{JKL}, \angle \mathrm{K}=64.3^{\circ}, k=22 \mathrm{~cm}, l=19 \mathrm{~cm} . </code> Find <code class='latex inline'>\displaystyle \angle \mathrm{L} </code>.</p>
<p> Determine whether the sine law or the cosine law should be used first to solve each triangle. Then, solve the triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree.</p><img src="/qimages/66379" />
<p>Sketch each triangle and label it with the given information. Then, use the given information to solve the triangle. Round your answers to the nearest tenth of a degree.</p><p>In acute <code class='latex inline'>\triangle NRX</code>, <code class='latex inline'>n = 7.4 km</code>, <code class='latex inline'>r = 8.3 km</code>, and <code class='latex inline'>x = 9.2 km</code>.</p>
<p>Find the length of x, to the nearest centimetre.</p><img src="/qimages/9471" />
<p>Suppose that the measured distances are taken between the centres of each pulley, and that the diameter of each pulley is <code class='latex inline'>2.8</code> cm. Find a more accurate total length of the drive belt. State any assumptions you make.</p><img src="/qimages/2527" />
<p>Find the indicated quantity, to the nearest tenth.</p><p>In <code class='latex inline'>\displaystyle \triangle \mathrm{DEF}, \angle \mathrm{D}=71.5^{\circ}, d=7.4 \mathrm{~m} </code>, and <code class='latex inline'>\displaystyle \angle \mathrm{E}=30.2^{\circ} . </code> Find <code class='latex inline'>\displaystyle f </code>.</p>
<p>Find the length of the indicated side in each triangle, to the nearest unit.</p><img src="/qimages/22909" />
<p>In an acute triangle, two sides are 2.4 cm and 3.6 cm. One of the angles is 37°. How can you determine the third side in the triangle? Explain.</p>
<p>Use the triangle at the right to create a problem that involves side lengths and interior angles. Then describe how to determine the length of side d.</p><img src="/qimages/1606" />
<p>Find <code class='latex inline'>\angle B</code>.</p><img src="/qimages/9444" />
<p>Sketch each triangle and use the given information to find the missing side length, to the nearest tenth of a unit.</p><p>In acute <code class='latex inline'>\triangle STU</code>, <code class='latex inline'>s = 1.3 mm</code>, <code class='latex inline'>u = 1.6 mm</code>, and <code class='latex inline'>\angle T = 49^{\circ}</code>.</p>
<p>Determine whether the sine law or the cosine law should be used first to solve each triangle. Then, solve the triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree.</p><img src="/qimages/66378" />
<p>Solve the triangle. Round answers to the nearest unit, if necessary.</p><img src="/qimages/9442" />
<p>Find the indicated angle in each triangle, to the nearest degree.</p><img src="/qimages/22921" />
<p>Solve each triangle. Round side lengths to the nearest tenth of a unit and angles to the nearest degree.</p><img src="/qimages/22911" />
<p>While flying at an altitude of <code class='latex inline'>1.5</code> km, a plane measures angles of depression to opposite ends of a large crater, as shown. Find the width of the crater, to the nearest tenth of a kilometre.</p><img src="/qimages/2547" />
<p>You and your partner are observing an aircraft from two observation decks, located <code class='latex inline'>5.0</code> km apart. From your point of view, the aircraft is at an angle of elevation of <code class='latex inline'>70</code><code class='latex inline'>^\circ</code>. </p><p>From your partner’s point of view, the angle of elevation is 55<code class='latex inline'>^\circ</code>. Determine the altitude of the aircraft, to the nearest tenth of a kilometre.</p><img src="/qimages/2514" />
<p> Use the cosine law to show that each angle of an equilateral triangle must measure <code class='latex inline'> 60^{\circ} </code> </p>
<p>Sketch each triangle and use the given information to find the missing side length, to the nearest tenth of a unit.</p><p>In acute <code class='latex inline'>\triangle PQR</code>, <code class='latex inline'>p = 1.8 m</code>, <code class='latex inline'>q = 2.1 m</code>, and <code class='latex inline'>\angle R = 73^{\circ}</code>.</p>
<p>Determine the indicated side length or angle measure in each triangle.</p><img src="/qimages/10071" />
<img src="/qimages/66332" />
<p>Solve each triangle. Round answers to the nearest unit, if necessary.</p><img src="/qimages/22903" />
<img src="/qimages/66327" />
<p><strong>a)</strong> Find <code class='latex inline'>x</code>, to the nearest tenth of a centimetre. </p><p><strong>b)</strong> Find <code class='latex inline'>x</code> using a different method. </p><img src="/qimages/2546" />
<p>Show that the cosine law simplifies to the Pythagorean theorem when the contained angle between the two known sides is 90<code class='latex inline'>^\circ</code></p>
<p>You and two of your team are to fly in a V-formation such that the distances between you are <code class='latex inline'>80</code> m, <code class='latex inline'>80</code> m, and <code class='latex inline'>100</code> m. Find the angles that illustrate how the three aircraft should be arranged, to the nearest tenth of a degree. Is there more than one possible solution? Explain.</p>
<p>Find the measure of the indicated angle in each triangle, to the nearest degree.</p><img src="/qimages/22898" />
<p> Given the side lengths in <code class='latex inline'> \triangle \mathrm{RST} </code> , find the area of the triangle, to the nearest square metre.</p><img src="/qimages/66367" />
<p>Lee is building a scale model of a water molecule for his science project. The molecule consists of one oxygen atom and two hydrogen atoms, chemically bonded as shown.</p><img src="/qimages/2529" /><p>Lee models the bond for each hydrogen atom with the oxygen atom using a 10-cm straw.</p><p><strong>a)</strong> How far will the two hydrogen atoms be from each other, to the nearest tenth of a centimetre?</p><p><strong>b)</strong> What angles will a line joining the two hydrogen atoms make with the lines of their chemical bonds?</p>
<p>Determine whether the primary trigonometric ratios, the sine law, or the cosine law should be used first to solve each triangle.</p><img src="/qimages/23046" />
<p>Sketch each triangle. Then. use the given information to find the indicated angle, to the nearest degree.</p><p><strong>a)</strong> In acute <code class='latex inline'>\triangle</code>ARD, <code class='latex inline'>a=170</code> mm, <code class='latex inline'>r=190</code> mm, and <code class='latex inline'>d=210</code> mm. Find <code class='latex inline'>\angle</code>D.</p><p><strong>b)</strong> In acute <code class='latex inline'>\triangle</code>HWN, <code class='latex inline'>h=1.4</code> km, <code class='latex inline'>w=1.7</code> km, and <code class='latex inline'>n=1.2</code> km. Find <code class='latex inline'>\angle</code>W.</p>
<p>Explain whether you can use the cosine law to find <code class='latex inline'> r </code> in <code class='latex inline'> \triangle </code> RST when given <code class='latex inline'> \angle T=53^{\circ}, t=15 \mathrm{~m} </code> , and <code class='latex inline'> s=14 \mathrm{~m} </code> .</p>
<p>Determine each unknown side length.</p><img src="/qimages/1601" />
<p>Find the indicated quantity, to the nearest tenth.</p><p>In acute <code class='latex inline'>\displaystyle \triangle \mathrm{PQR}, \angle \mathrm{Q}=53.7^{\circ}, q=35 \mathrm{~m}, r=28 \mathrm{~m} </code>. Find <code class='latex inline'>\displaystyle \angle \mathrm{P} </code></p>
<p>Find the indicated quantity, to the nearest tenth.</p><p>In <code class='latex inline'>\displaystyle \triangle \mathrm{GHK}, \angle \mathrm{G}=44.1^{\circ}, k=9.5 \mathrm{~cm} </code>, and <code class='latex inline'>\displaystyle \angle \mathrm{H}=78.4^{\circ} </code>. Find <code class='latex inline'>\displaystyle h </code>.</p>
<p>A ship is sighted at sea from two observation points on the coastline that are 60 km apart. The angle between the coastline and the ship at the first observation point is 43°. From the second observation point, the angle between the coastline and the ship is55°.Howfaristheshipfromthe second observation point, to the nearest kilometre?</p><img src="/qimages/9472" />
<p>Taylor and Brooklyn are recording how far a ball rolls down a ramp during each second. The table below shows the data they have collected.</p><p>Write an equation for the sequence.</p>
<p>Determine the measure of each indicated angle to the nearest degree,</p><img src="/qimages/1603" />
<p>Determine the perimeter <code class='latex inline'>\triangle SRT</code>, if <code class='latex inline'>\angle S</code> = 60°, <code class='latex inline'>r = 15 cm</code>, and <code class='latex inline'>t = 20 cm</code>.</p>
<p>Find the measure of the indicated angle in each triangle, to the nearest degree.</p><p><strong>a)</strong> </p><img src="/qimages/2505" /><p><strong>b)</strong></p><img src="/qimages/2506" />
<p>Three checkpoints, A, B, and C, have been set up in a park for a fundraising relay run.</p><img src="/qimages/22931" /><p>Find the measure of each of the angles, to the nearest degree, inside the triangle that connects the three checkpoints.</p>
<p>Sketch each triangle and label the given information. Then, use the given information to find the indicated angle, to the nearest degree.</p><p>In acute <code class='latex inline'>\triangle ABC</code>, <code class='latex inline'>a = 3.5 cm</code>, <code class='latex inline'>b = 4.1 cm</code>, and <code class='latex inline'>c = 5.4 cm</code>. Find <code class='latex inline'>\angle A</code>.</p>
<p>Solve each triangle. Round each calculated value to the nearest tenth of a unit, if necessary.</p><img src="/qimages/66353" />
<p> Try to solve a triangle with side lengths <code class='latex inline'> 3 \mathrm{~cm}, 8 \mathrm{~cm} </code> , and <code class='latex inline'> 4 \mathrm{~cm} </code> using the cosine law. What do you find? Explain why.</p>
<p>Solve the triangle. Round answers to the nearest unit, if necessary.</p><img src="/qimages/9472" />
<p>Find the length of the indicated side in each triangle, to the nearest unit.</p><img src="/qimages/22910" />
<p>Find the length of the walkway across the river, to the nearest metre.</p><img src="/qimages/22915" />
<p>Leia is in a bicycle road race. In the first leg, she rides <code class='latex inline'>12</code> km from Riverside to Danton. Then, she turns and rides 17 km to Humberville, making a 74<code class='latex inline'>^\circ</code> angle from the first leg. The final turn leads back to Riverside.</p><p><strong>a)</strong> What is the total length of the race, to the nearest kilometre?</p><p><strong>b)</strong> At what angles are the three towns situated with respect to each other? Round to the nearest degree.</p>
<p>Hanna, Jon. and Robin live in two identical apartment buildings. located <code class='latex inline'>30</code> in apart. Jon lives two floors higher than Hanna. Robin lives four floors lower than Hanna. There is a 36<code class='latex inline'>^\circ</code> angle of separation when Hanna looks from her balcony to those of her two friends.</p><img src="/qimages/2554" /><p><strong>a)</strong> How far apart, vertically, do Jon and Robin live? Round to the nearest tenth of a metre.</p><p><strong>b)</strong> Explain how you solved this problem and discuss any assumptions you made.</p>
<img src="/qimages/10081" /><p>What is the measure of <code class='latex inline'>\displaystyle \theta </code> ?</p><p>A. <code class='latex inline'>\displaystyle 13^{\circ} </code></p><p>B. <code class='latex inline'>\displaystyle 43^{\circ} </code></p><p>C. <code class='latex inline'>\displaystyle 73^{\circ} </code></p><p>D. <code class='latex inline'>\displaystyle 47^{\circ} </code></p>
<p>Find the length of <code class='latex inline'>x</code>. to the nearest centimetre.</p><img src="/qimages/5709" />
<p> Solve each triangle. Round each calculated value to the nearest tenth of a unit, if necessary.</p><p>In <code class='latex inline'>\displaystyle \triangle \mathrm{TUW}, w=25.4 \mathrm{~cm}, u=34.2 \mathrm{~cm} </code>, and <code class='latex inline'>\displaystyle \angle \mathrm{T}=43.1^{\circ} </code></p>
<p>Determine whether the sine law or the cosine law should be used first to solve each triangle. Then, solve the triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree.</p><img src="/qimages/66380" />
<p>Determine each unknown side length.</p><img src="/qimages/1602" />
<p>Find the length of the indicated side in each triangle, to the nearest unit.</p><img src="/qimages/22892" />
<p>Find the missing side length in each triangle, to the nearest tenth of a unit.</p><p><strong>a)</strong></p><img src="/qimages/2520" /><p><strong>b)</strong></p><img src="/qimages/2519" /><p><strong>c)</strong></p><img src="/qimages/2521" />
<p>Solve the following triangles. Round lengths to the nearest tenth of a metre and angle measures to the nearest degree.</p><p>In <code class='latex inline'>\triangle ABC, \angle A =68^o</code>, <code class='latex inline'>b= 5 cm</code> and <code class='latex inline'>c =7 cm</code>.</p>
<p>Solve each triangle. Round answers to the nearest unit, if necessary. </p><p>a) </p><img src="/qimages/2507" /><p>b) </p><img src="/qimages/2508" />
<p>Solve <code class='latex inline'>\triangle ABC</code>, if <code class='latex inline'>\angle A = 75^o</code>, <code class='latex inline'>\angle B = 50^o</code>, and the side between these angles is <code class='latex inline'>8.0 cm</code>.</p>
<p>Doctors Jones and Hwang are astronomers observing the sun from opposite ends of Earth. The radius of Earth is 6400 km.</p><img src="/qimages/2552" /><p><strong>a)</strong> Use this information to verify the distance from Earth to the Sun, which was given in question 5. State any assumptions you make.</p><p><strong>b)</strong> At approximately what times of day were these observations made by each astronomer? Explain your answer.</p>
<p>Sketch each triangle and label it with the given information. Then, use the given information to solve the triangle. Round your answers to the nearest tenth of a degree.</p><p>In acute <code class='latex inline'>\triangle KQV</code>, <code class='latex inline'>k = 15.7 m</code>, <code class='latex inline'>q = 16.8 m</code>, and <code class='latex inline'>v = 17.3 m</code>.</p>
<p>Acute <code class='latex inline'>\triangle ABC</code> is scalene with <code class='latex inline'>b = 2a</code>. Show that <code class='latex inline'>\cos C = \dfrac{5}{4} - \dfrac{c^2}{4a^2}</code>.</p>
<p> Solve each triangle. Round each calculated value to the nearest tenth of a unit, if necessary.</p><p>In <code class='latex inline'>\displaystyle \triangle \mathrm{IJK}, i=10.5 \mathrm{~m}, j=11.8 \mathrm{~m} </code>, and <code class='latex inline'>\displaystyle k=12.5 \mathrm{~m} </code>.</p>
<ul> <li>a) Determine the length of side <code class='latex inline'>x</code>.</li> <li>b) Determine the measure of <code class='latex inline'>\angle P</code>.</li> </ul> <img src="/qimages/1600" />
<p>A parallelogram has sides that are 8 cm and 15 cm long. One of the angles in the parallelogram measures 70°. Explain how you could calculate the length of the shortest diagonal.</p>
<p>he three sides of a triangle measure 15 cm, 17 cm, and 18 cm. Find the measure of the largest angle, to the nearest degree.</p>
<p>An airplane is flying from Montreal to Vancouver. The wind is blowing from the west at 60 km/h. The airplane flies at an airspeed of 750 km/h and must stay on a heading of65° west of north.</p> <ul> <li>What is the speed of the airplane relative to the ground?</li> </ul>
<p> Solve <code class='latex inline'>\displaystyle \triangle \mathrm{ABC} </code> with vertices <code class='latex inline'>\displaystyle \mathrm{A}(2,1), \mathrm{B}(7,4) </code>, and <code class='latex inline'>\displaystyle \mathrm{C}(1,5) </code>. Round each calculated value to the nearest tenth of a unit.</p>
<p>Solve the triangle. Round answers to the nearest tenth of a degree.</p><p>In acute <code class='latex inline'>\triangle VSF</code>, <code class='latex inline'>v = 2.9 km, s =3.5 km</code>, and <code class='latex inline'>f = 3.0 km</code>.</p>
<p>Find the length <code class='latex inline'>x</code>, to the nearest tenth of a metre.</p><img src="/qimages/9440" />
<p>Sketch each triangle and use the given information to find the missing side length, to the nearest tenth of a unit.</p><p>In acute <code class='latex inline'>\triangle VWX</code>, <code class='latex inline'>w = 7.2 km</code>, <code class='latex inline'>x = 5.3 km</code>, and <code class='latex inline'>\angle V = 52^{\circ}</code>.</p>
<p><strong>a)</strong> Show that the sine law simplifies to the sine ratio when one of the angles of a triangle is <code class='latex inline'>90</code><code class='latex inline'>^\circ</code>.</p><p><strong>b)</strong> Would you use the sine law to solve right triangles? Explain.</p>
<p>Solve each triangle. Round side lengths to the nearest tenth of a unit and angles to the nearest degree.</p><img src="/qimages/22912" />
<p>Solve the following triangles. Round lengths to the nearest tenth of a metre and angle measures to the nearest degree.</p><p>In <code class='latex inline'>\triangle RST, \angle R = 73^o, r = 8m</code> and <code class='latex inline'>t = 6 m</code>.</p>
<p>Sketch each triangle and label the given information. Then, solve the triangle. Round side lengths to the nearest tenth of a unit and angles to the nearest degree.</p><p>In <code class='latex inline'>\triangle TYH</code>, <code class='latex inline'>\angle T = 61^{\circ}</code>, <code class='latex inline'>y = 14 cm</code>, and <code class='latex inline'>h= 19 cm</code>.</p>
<p>Find the indicated angle in each triangle, to the nearest degree.</p><img src="/qimages/22922" />
<p>Solve each of the following acute triangles. Round each answer to the nearest whole number, if necessary.</p><img src="/qimages/66321" />
<p>Sketch each triangle and label the given information. Then, use the given information to find the indicated angle, to the nearest degree.</p><p>In acute <code class='latex inline'>\triangle DEF</code>, <code class='latex inline'>d = 14.1 km</code>, <code class='latex inline'>e = 15.2 km</code>, and <code class='latex inline'>f = 16.3 km</code>. Find <code class='latex inline'>\angle D</code>.</p>
<p>Sketch each triangle and use the given information to find the missing side length, to the nearest tenth of a unit.</p><p><strong>a)</strong> In acute <code class='latex inline'>\triangle</code>TUV, <code class='latex inline'>t=1.8</code>cm, <code class='latex inline'>v=1.4</code>cm, and <code class='latex inline'>\angle</code>U = 52<code class='latex inline'>^\circ</code></p><p><strong>b)</strong> In acute <code class='latex inline'>\triangle</code>DEF, <code class='latex inline'>e=1.1</code>km, <code class='latex inline'>f=1.6</code>km, and <code class='latex inline'>\angle</code>D = 74<code class='latex inline'>^\circ</code></p>
<p>Sketch each triangle and label the given information. Then, use the given information to find the indicated angle, to the nearest degree.</p><p>In acute <code class='latex inline'>\triangle PQR</code>, <code class='latex inline'>p = 11.3 m</code>, <code class='latex inline'>q = 12.4 m</code>, and <code class='latex inline'>r = 13.2 m</code>. Find <code class='latex inline'>\angle Q</code>.</p>
<p>The airport at Winnipeg Manitoba, has two runways with lengths <code class='latex inline'>1525</code> m and <code class='latex inline'>915</code> m. The beginnings of the runways meet at an angle of <code class='latex inline'>37</code><code class='latex inline'>^\circ</code>. The other ends of the runways are called the thresholds.</p><p>a) Draw a diagram and label the given information.</p><p>b) How far apart are the thresholds of the runways, to the nearest metre?</p>
<p>Find the indicated angle, to the nearest degree.</p><img src="/qimages/66341" />
<p>Solve each triangle. Round answers to the nearest unit, if necessary.</p><img src="/qimages/5711" />
<p>Find the indicated quantity, to the nearest tenth of a unit.</p><p>In <code class='latex inline'>\displaystyle \triangle \mathrm{PQR}, p=15.3 \mathrm{~m}, q=18.2 \mathrm{~m} </code>, and <code class='latex inline'>\displaystyle \angle \mathrm{R}=70.5^{\circ} </code>. Find <code class='latex inline'>\displaystyle r </code>.</p>
<p>Solve the triangle. Round each side length to the nearest tenth of a unit and each angle to the nearest degree.</p><img src="/qimages/9468" />
<p>Trey, who is <code class='latex inline'>1.5</code> m tall, is standing at a distance of <code class='latex inline'>14</code> m from a building. From his point of view, the bottom and top of the building are separated by 36<code class='latex inline'>^\circ</code>, as shown. How tall is the building, to the nearest tenth of a metre?</p><img src="/qimages/2548" />
<p>Sketch each triangle and label the given information. Then, solve the triangle. Round side lengths to the nearest tenth of a unit and angles to the nearest degree.</p><p>In <code class='latex inline'>\triangle JRC</code>, <code class='latex inline'>\angle R = 48^{\circ}</code>, <code class='latex inline'>j = 15.2 mm</code>, and <code class='latex inline'>c= 17.3 mm</code>.</p>
<p>Find the length of the indicated side, to the nearest metre.</p><img src="/qimages/66285" />
<p>Find the length of <code class='latex inline'>c</code>, to the nearest tenth of a metre.</p><img src="/qimages/9472" />
<p>Find the length of the indicated side in each triangle, to the nearest unit.</p><img src="/qimages/22895" />
<p> Solve each of the following triangles. Round each answer to the nearest whole number, if necessary.</p><img src="/qimages/66315" />
<p>Draw a diagram and label the given information. Then, solve each triangle. Round answers to the nearest unit, if necessary. </p><p><strong>a)</strong> In acute <code class='latex inline'>\triangle</code>AKR, <code class='latex inline'>k=15</code>mm, <code class='latex inline'>r=13</code>mm, and <code class='latex inline'>\angle</code>K = 68<code class='latex inline'>^\circ</code></p><p><strong>b)</strong> In acute <code class='latex inline'>\triangle</code>UJF, <code class='latex inline'>j=23</code>km, <code class='latex inline'>\angle</code>U = 57<code class='latex inline'>^\circ</code>, and <code class='latex inline'>\angle</code>F = 48<code class='latex inline'>^\circ</code></p>
<p>Determine whether the primary trigonometric ratios, the sine law, or the cosine law should be used first to solve each triangle. </p><p><strong>a)</strong></p><img src="/qimages/2542" /><p><strong>b)</strong></p><img src="/qimages/2543" /><p><strong>c)</strong></p><img src="/qimages/2544" /><p><strong>d)</strong></p><img src="/qimages/2545" />
<p>Draw a diagram and label the given information. Then. solve each triangle Round answers to the nearest unit. if necessary.</p><p><strong>a)</strong> In acute <code class='latex inline'>\triangle</code>EFG, <code class='latex inline'>e=5</code>cm, <code class='latex inline'>f=6</code>cm, and <code class='latex inline'>\angle</code>G = 63<code class='latex inline'>^\circ</code></p><p><strong>b)</strong> In acute <code class='latex inline'>\triangle</code>WXY, <code class='latex inline'>w=10</code>m, <code class='latex inline'>y=11</code>m, and <code class='latex inline'>\angle</code>X = 80<code class='latex inline'>^\circ</code></p>
<p>The angle between two equal sides of an isosceles triangle is <code class='latex inline'>52^o</code>.</p><p>Each of the equal sides is 18 cm long.</p> <ul> <li>Determine the measures of the two equal angles in the triangle.</li> </ul>
<p>Find the length of the indicated side in each triangle, to the nearest unit. </p><p><strong>a)</strong> </p><img src="/qimages/2501" /><p><strong>b)</strong></p><img src="/qimages/2502" />
<p>Solve <code class='latex inline'>\triangle XYZ</code>. Round the side length to the nearest tenth of a metre and the angle measures to the nearest degree.</p><img src="/qimages/9451" />
<p>The angle between two equal sides of an isosceles triangle is <code class='latex inline'>52^o</code>.</p><p>Each of the equal sides is 18 cm long.</p> <ul> <li>Determine the perimeter of the triangle.</li> </ul>
<p> The tops of the solid rocket boosters used to launch the space shuttle are cones of diameter <code class='latex inline'> 3.7 \mathrm{~m} </code> and slant height <code class='latex inline'> 5.4 \mathrm{~m} </code> . Find the angle that the curved surface of the cone makes with a diameter, to the nearest tenth of a degree.</p><img src="/qimages/66361" />
<p>In isosceles <code class='latex inline'>\triangle</code>ABC, <code class='latex inline'>c=15</code>cm, <code class='latex inline'>a=11</code>cm, and <code class='latex inline'>\angle</code>B = 68.5<code class='latex inline'>^\circ</code>. Can this triangle be solved using the sine law? If so, solve it and explain your reasoning. If not, explain why not.</p>
<p>Draw a diagram and label the given information. Then, find the measure of the indicated side in each triangle, to the nearest tenth of a unit.</p><p>In acute <code class='latex inline'>\triangle PQR</code>, <code class='latex inline'>\angle P = 68^{\circ}</code>, <code class='latex inline'>\angle R = 51^{\circ}</code>, and <code class='latex inline'>r = 18.2 cm</code>. Find side <code class='latex inline'>p</code>.</p>
<p>Calculate the indicated side length or angle measure in each triangle.</p><img src="/qimages/9940" />
<p> Explain whether you can use the sine law to find <code class='latex inline'> p </code> in <code class='latex inline'> \triangle P Q R </code> when given <code class='latex inline'> \angle P=47^{\circ}, q=9 \mathrm{~m} </code> , and <code class='latex inline'> r=11 \mathrm{~m} </code> .</p>
<p>Find the length of the indicated side, to the nearest metre.</p><img src="/qimages/66288" />
<p>Solve the triangle.</p> <ul> <li>In <code class='latex inline'>\triangle PQR, p=6.4m, q=9.0 cm</code>, and <code class='latex inline'>\angle R</code>=80°.</li> </ul>
<p>Find the missing side length, to the nearest tenth of a unit.</p><img src="/qimages/66337" />
<p>A regular pentagon is inscribed in a circle with radius 10 cm as shown in the diagram below. Determine the perimeter of the pentagon.</p><img src="/qimages/1612" />
<p>A ship travels 100 km at a bearing of N60<code class='latex inline'>^\circ</code>E and then turns and travels 80 km at a bearing of S2O<code class='latex inline'>^\circ</code>E before reaching its destination. Suppose the ship travelled directly from its starting point to its destination, following a direct route. What distance and at what bearing would the ship travel? Round to the nearest unit.</p>
<p>Find the indicated angle, to the nearest degree.</p><img src="/qimages/66342" />
<p>Solve each triangle. Round answers to the nearest unit, if necessary.</p><p><strong>a)</strong></p><img src="/qimages/2509" /><p><strong>b)</strong></p><img src="/qimages/2510" />
<p>Find <code class='latex inline'>b</code>.</p><img src="/qimages/9445" />
<p>An awning over a window is supported by a cable attached to a building, as shown. The cable is attached to the building 2.1 m above the base of the awning.</p><img src="/qimages/22905" /><p>a) Find the length of the cable, to the nearest tenth of a metre.</p><p>b) Find the length of the awning, to the nearest tenth of a metre.</p>
<p>Solve the following triangles. Round lengths to the nearest tenth of a metre and angle measures to the nearest degree.</p><p>In <code class='latex inline'>\triangle DEF, \angle D = 58^o, \angle E = 53^o</code>, and <code class='latex inline'>d = 8 cm</code>.</p>
<p>Determine the measure of <code class='latex inline'>\displaystyle \angle P </code>.</p><img src="/qimages/156256" />
<p>Solve each triangle. Round your answers to the nearest tenth of a degree.</p><img src="/qimages/22924" />
<p>A tetrahedron has edges that are <code class='latex inline'>10</code> cm in length. Find the height of this tetrahedron. to the nearest tenth of a centimetre.</p><img src="/qimages/2551" />
<p>Solve the triangle. Round answers to the nearest unit, if necessary.</p><p>In acute <code class='latex inline'>\triangle EFG, \angle E = 84^o</code>, <code class='latex inline'>f = 32 km</code>, and <code class='latex inline'>g = 21 km</code>.</p>
<p>The longest side of a triangle is <code class='latex inline'> 50 \mathrm{~cm} </code> . The measures of two angles in the triangle are <code class='latex inline'> 42^{\circ} </code> and <code class='latex inline'> 64^{\circ} </code> . Find the lengths of the other two sides, to the nearest centimetre.</p>
<p>For each of the following acute triangles, find the measure of the indicated angle, to the nearest degree.</p><img src="/qimages/66294" />
<p>Find the length of the indicated side in each triangle, to the nearest unit.</p><img src="/qimages/22894" />
<p>Draw a diagram and label the given information. Then, find the measure of the indicated side in each triangle, to the nearest tenth of a unit.</p><p>In acute <code class='latex inline'>\triangle GHI</code>, <code class='latex inline'>\angle G = 80^{\circ}</code>, <code class='latex inline'>\angle I = 37^{\circ}</code>, and <code class='latex inline'>g = 10 m</code>. Find side <code class='latex inline'>i</code>.</p>
<p>Find the indicated quantity, to the nearest tenth of a unit.</p><p>In <code class='latex inline'>\displaystyle \triangle \mathrm{BCD}, b=10.8 \mathrm{~cm}, c=22.1 \mathrm{~cm} </code>, and <code class='latex inline'>\displaystyle d=22.1 \mathrm{~cm} </code>. Find <code class='latex inline'>\displaystyle \angle \mathrm{B} </code>.</p>
<p>Sketch each triangle and label the given information. Then, solve the triangle. Round side lengths to the nearest tenth of a unit and angles to the nearest degree.</p><p>In <code class='latex inline'>\triangle FYZ</code>, <code class='latex inline'>\angle Y = 61^{\circ}</code>, <code class='latex inline'>f = 8.3 km</code>, and <code class='latex inline'>z= 5.4 km</code>.</p>
<p>Sally makes stained glass windows. Each piece of glass is surrounded by lead edging. Sally claims that she can create an acute triangle in part of a window using pieces of lead that are 15 cm, 36 cm, and 60 cm.</p><p>Is she correct? Justify your decision.</p>
<p>Find the measure of the indicated angle, to the nearest degree.</p><img src="/qimages/9448" />
<p>Find the length of the indicated side in each triangle, to the nearest unit.</p><img src="/qimages/22907" />
<p>Solve the triangle. Round answers to the nearest unit, if necessary.</p><p>In acute <code class='latex inline'>\triangle WXY, \angle X = 81^o</code>, <code class='latex inline'>\angle W = 32^o</code>, and <code class='latex inline'>w = 16 cm</code>.</p>
<p> Solve each of the following acute triangles. Round each answer to the nearest whole number, if necessary.</p><img src="/qimages/66320" />
<p>Find the length of the indicated side in each triangle, to the nearest tenth of a unit.</p><p><strong>a)</strong></p><img src="/qimages/2503" /><p><strong>b)</strong></p><img src="/qimages/2504" />
<p>Solve the triangle. Round each side length to the nearest tenth of a unit and each angle to the nearest degree.</p><img src="/qimages/9469" />
<p>Solve each triangle. Round answers to the nearest unit, if necessary.</p><img src="/qimages/22904" />
<p>A hydro pole is supported by a 20-m guy wire that makes an angle of <code class='latex inline'>48^{\circ}</code> with the horizontal ground. A 16-m guy wire is to be fastened on the other side of the pole for reinforcement. Both wires attach to the pole at its top.</p><p>a) What angle should the second wire make with the ground? Round your answer to the nearest degree.</p><p>b) How tall is the hydro pole? Round your answer to the nearest tenth of a metre.</p><p>c) How far is the base of the 20-m guy wire from the base of the 16-m guy wire? Round your answer to the nearest tenth of a metre.</p>
<p>Draw a diagram and label the given information. Then, solve each triangle. Round answers to the nearest unit, if necessary.</p><p>In <code class='latex inline'>\triangle KPR</code>, <code class='latex inline'>\angle K = 63^{\circ}</code>, <code class='latex inline'>\angle P = 71^{\circ}</code>, and <code class='latex inline'>r = 13 m</code>.</p>
<p>Suppose that you are given each set of data for <code class='latex inline'>\triangle ABC</code> at the right. Can you use the cosine law to determine <code class='latex inline'>c</code>? Explain.</p> <ul> <li><code class='latex inline'>a = 5cm, b = 7 cm, \angle C = 43^o</code></li> </ul> <img src="/qimages/1599" />
<p>Solve each of the following triangles. Round each answer to the nearest whole number, if necessary.</p><img src="/qimages/66313" />
<p>In <code class='latex inline'>\triangle XYZ</code>, <code class='latex inline'>\angle X =</code> 90°, <code class='latex inline'>XY = 3.5 cm</code>, and <code class='latex inline'>YZ = 4.8 cm</code>. Solve <code class='latex inline'>\triangle XYZ</code>. Round lengths to the nearest tenth of a metre and angle measures to the newest degree.</p>
<p>Determine the perimeter of isosceles <code class='latex inline'>\triangle ABC</code>, to the nearest tenth of a centimetre.</p><img src="/qimages/9439" />
<p>Solve each triangle. Round side lengths to the nearest tenth of a unit and angles to the nearest degree.</p><img src="/qimages/22913" />
<p>Draw an equilateral triangle and label its sides <code class='latex inline'>a</code>. Mark one of the angles 60<code class='latex inline'>^\circ</code>. Use the cosine law to prove that cos 60<code class='latex inline'>\displaystyle{^\circ=\frac{1}{2}}</code></p>
<p>Solve each triangle. Round each calculated value to the nearest tenth of a unit, if necessary.</p><img src="/qimages/66354" />
<p>Calculate the indicated side length or angle measure in each triangle.</p><img src="/qimages/9939" />
<p>Consider <code class='latex inline'>\triangle</code>JVM.</p><p><strong>a)</strong> Follow these steps in order to solve <code class='latex inline'>\triangle</code>JVM. Round answers to the nearest tenth of a degree.</p> <ul> <li><p>Use the cosine law to find <code class='latex inline'>\angle</code>J. </p></li> <li><p>Use the cosine law to find <code class='latex inline'>\angle</code>V. </p></li> <li><p>Use the cosine law to find <code class='latex inline'>\angle</code>M.</p></li> </ul> <p><strong>b)</strong> Solve <code class='latex inline'>\triangle</code>JVM using a more efficient method. </p><p><strong>c)</strong> Compare your answers in parts a) and b). Explain why your method is more efficient.</p>
<p>Sketch each triangle and label it with the given information. Then, use the given information to solve the triangle. Round your answers to the nearest tenth of a degree.</p><p>In acute <code class='latex inline'>\triangle LWG</code>, <code class='latex inline'>l = 8.2 cm</code>, <code class='latex inline'>w = 9.3 cm</code>, and <code class='latex inline'>g = 10.4 cm</code>.</p>
<p>Find the indicated quantity, to the nearest tenth.</p><p>In <code class='latex inline'>\displaystyle \triangle \mathrm{ABC}, \angle \mathrm{B}=38.2^{\circ}, \angle \mathrm{C}=65.6^{\circ} </code>, and <code class='latex inline'>\displaystyle b=54 \mathrm{~cm} . </code> Find <code class='latex inline'>\displaystyle c </code>.</p>
<p>Find the missing side length, to the nearest tenth of a unit.</p><img src="/qimages/66336" />
<p>A leaning pole is braced at its midpoint, as shown. The pole is <code class='latex inline'>8.2</code> m long, and the bracing beam is <code class='latex inline'>6.0</code> m long. The foot of the beam is placed <code class='latex inline'>5.0</code> m from the base of the pole. Determine, to the nearest degree,</p><p><strong>a)</strong> the angle the pole makes with the ground</p><p><strong>b)</strong> the angle the beam makes with the ground</p><p><strong>c)</strong> the angle the beam makes with the pole</p>
<p>Find the indicated quantity, to the nearest tenth of a unit.</p><p>In <code class='latex inline'>\displaystyle \triangle \mathrm{UVW}, u=10.3 \mathrm{~cm}, v=11.4 \mathrm{~cm} </code>, and <code class='latex inline'>\displaystyle w=12.5 \mathrm{~cm} </code>. Find <code class='latex inline'>\displaystyle \angle \mathrm{V} </code></p>
<p>Suppose that you are given each set of data for <code class='latex inline'>\triangle ABC</code> at the right. Can you use the cosine law to determine <code class='latex inline'>c</code>? Explain.</p> <ul> <li><code class='latex inline'>a = 5cm, \angle A = 52^o, \angle C = 43^o</code></li> </ul> <img src="/qimages/1599" />
<p>Becky is standing on an observation deck in the forest. From where she is standing she can see her daughters Lauren and Clara. She estimates how far away they are from her and the angle separating their lines of sight, as shown.</p><img src="/qimages/22916" /><p>Use Becky&#39;s estimated measures.</p><p>a) How far apart are the Lauren and Clara, to the nearest tenth of a metre?</p><p>b) At what angle of elevation does Becky appear to Lauren, to the nearest degree?</p><p>c) At what angle of elevation does Becky appear to Clara, to the nearest degree?</p>
<p>Draw a diagram and label the given information. Then, find the measure of the indicated side in each triangle, to the nearest degree.</p><p>In acute <code class='latex inline'>\triangle XYZ</code>, <code class='latex inline'>\angle X = 64^{\circ}</code>, <code class='latex inline'>x = 10.4 cm</code>, and <code class='latex inline'>y = 7.1 cm</code>. Find <code class='latex inline'>\angle Y</code>.</p>
<p> For each of the following acute triangles, find the measure of the indicated angle, to the nearest degree.</p><img src="/qimages/66295" />
<p>Find the indicated angle in each triangle, to the nearest degree.</p><img src="/qimages/22920" />
<p>Find the indicated quantity, to the nearest tenth.</p><p>In acute <code class='latex inline'>\displaystyle \triangle \mathrm{RST}, \angle \mathrm{T}=56.5^{\circ}, t=8.2 \mathrm{~m}, r=9.3 \mathrm{~m} </code>. Find <code class='latex inline'>\displaystyle \angle \mathrm{R} </code>.</p>
<p>Sketch each triangle and label the given information. Then, use the given information to find the indicated angle, to the nearest degree.</p><p>In acute <code class='latex inline'>\triangle XYZ</code>, <code class='latex inline'>x = 21.6 mm</code>, <code class='latex inline'>y = 23.4 mm</code>, and <code class='latex inline'>z = 24.5 mm</code>. Find <code class='latex inline'>\angle X</code>.</p>
<p>In <code class='latex inline'>\triangle PQR, \angle P = 80^o, \angle Q = 48^o</code>, and <code class='latex inline'>r = 20 cm</code>. Solve <code class='latex inline'>\triangle PQR</code>.</p>
<p>The angle between two equal sides of an isosceles triangle is <code class='latex inline'>52^o</code>.</p><p>Each of the equal sides is 18 cm long.</p> <ul> <li>Determine the length of the third side.</li> </ul>
<p>The distance from the centre, 0, of a regular decagon to each vertex is 12 cm. Calculate the area of the decagon.</p><img src="/qimages/1605" />
<p>The pendulum of a grandfather clock is <code class='latex inline'>100.0 cm</code> long. When the pendulum swings from one side to the other side, the horizontal distance it travels is <code class='latex inline'>9.6 cm</code>, as in the diagram at the right. Determine the angle through which the pendulum swings. Round your answer to the nearest tenth of a degree.</p><img src="/qimages/1604" />
<p>Solve each triangle. Round your answers to the nearest tenth of a degree.</p><img src="/qimages/22926" />
<img src="/qimages/10083" /><p>Which statements are true for <code class='latex inline'>\displaystyle \triangle A B C ? </code></p><p>A. <code class='latex inline'>\displaystyle \frac{a}{\sin A}=\frac{b}{\sin B} </code> and <code class='latex inline'>\displaystyle a^{2}=b^{2}+c^{2}-2 b c \cos A </code></p><p>B. <code class='latex inline'>\displaystyle \frac{a}{\sin A}=\frac{\sin B}{b} </code> and <code class='latex inline'>\displaystyle a^{2}=b^{2}+c^{2}-2 b c \cos A </code></p><p>C. <code class='latex inline'>\displaystyle \frac{a}{\sin A}=\frac{b}{\sin B} </code> and <code class='latex inline'>\displaystyle a^{2}=b^{2}+c^{2}+2 b c \cos A </code></p><p>D. <code class='latex inline'>\displaystyle \frac{a}{\sin A}=\frac{b}{\sin B} </code> and <code class='latex inline'>\displaystyle c^{2}=a^{2}+b^{2}-2 a b \cos A </code></p>
<p>Sketch each triangle and label it with the given information. Then. solve the triangle. Round answers to the nearest tenth of a degree.</p><p><strong>a)</strong> In acute <code class='latex inline'>\triangle</code>NBG, <code class='latex inline'>n=15</code> m, <code class='latex inline'>b=14</code> m and <code class='latex inline'>g=12</code> m.</p><p><strong>b)</strong> In acute <code class='latex inline'>\triangle</code>DRT, <code class='latex inline'>d=5.0</code> km, <code class='latex inline'>r=3.8</code> km and <code class='latex inline'>t=4.6</code> km.</p>
<img src="/qimages/66333" />
<p>Acute <code class='latex inline'>\triangle ABC</code> is scalene with <code class='latex inline'>c = 3b</code>. Show that <code class='latex inline'>cos A = \dfrac{5}{3} - \dfrac{a^2}{6b^2}</code>.</p>
<p>Calculate the perimeter and area of this regular pentagon. O is the centre of this pentagon.</p><img src="/qimages/1607" />
<p>In acute <code class='latex inline'>\triangle EQW</code>, <code class='latex inline'>AW =</code> 77°, <code class='latex inline'>e = 11 km</code>, and <code class='latex inline'>q = 14km</code>.</p>
<p>The bases in a baseball diamond are 90 ft apart. A player picks up a ground ball 1 1 ft from third base, along the line from second base to third base. Determine the angle that is formed between first base, the players present position, and home plate.</p>
<img src="/qimages/10072" /><p>Which triangle is similar to <code class='latex inline'>\displaystyle \triangle A B C </code> ?</p><p>A. <code class='latex inline'>\displaystyle \triangle D E F </code>, with <code class='latex inline'>\displaystyle \angle D=45^{\circ}, \angle E=62^{\circ} </code></p><p>B. <code class='latex inline'>\displaystyle \triangle D E F </code>, with <code class='latex inline'>\displaystyle f=12.5, d=9.0 \mathrm{~cm} </code>, <code class='latex inline'>\displaystyle e=10.5 \mathrm{~cm} </code></p><p>C. <code class='latex inline'>\displaystyle \triangle D E F </code>, with <code class='latex inline'>\displaystyle \angle F=79^{\circ}, d=7.2 \mathrm{~cm} </code>, <code class='latex inline'>\displaystyle e=12.6 \mathrm{~cm} </code></p><p>D. <code class='latex inline'>\displaystyle \triangle D E F </code>, with <code class='latex inline'>\displaystyle f=10 \mathrm{~cm}, d=10.8 \mathrm{~cm} </code>, <code class='latex inline'>\displaystyle \angle E=56^{\circ} </code></p>
<p>Solve the triangle. Round answers to the nearest tenth of a degree.</p><p>In acute <code class='latex inline'>\triangle ABC, a = 6.8 cm</code>, <code class='latex inline'>b = 8.7 cm</code>, and <code class='latex inline'>c = 9.6 cm</code>.</p>
<p>Dominique and Allison are observing a hot-air balloon from two tracking stations on the ground. The tracking stations are 6.0 km apart. From Dominique’s point of view, the hot-air balloon is at an angle of elevation of <code class='latex inline'>72^{\circ}</code>. From Allison’s point of view, the angle of elevation is <code class='latex inline'>58^{\circ}</code>.</p><img src="/qimages/22906" /><p>a) Determine the distance that Dominique is from the hot-air balloon, to the nearest tenth of a kilometre.</p><p>b) Determine the distance that Allison is from the hot-air balloon, to the nearest tenth of a kilometre.</p><p>C) Determine the altitude of the hot-air balloon, to the nearest tenth of a kilometre.</p>
<p> Solve each of the following acute triangles. Round each answer to the nearest whole number, if necessary.</p><img src="/qimages/66319" />
<p>Solve each triangle. Round your answers to the nearest tenth of a degree.</p><img src="/qimages/22925" />
<p>Draw a diagram and label the given information. Then, find the measure of the indicated angle in each triangle, to the nearest degree. </p><p>a) In acute <code class='latex inline'>\triangle</code>GHK, <code class='latex inline'>\angle</code>G = 47<code class='latex inline'>^\circ</code>, <code class='latex inline'>h=5</code>cm, and <code class='latex inline'>g=4</code>cm. Find <code class='latex inline'>\angle</code>H.</p><p>b) In acute <code class='latex inline'>\triangle</code>RST, <code class='latex inline'>\angle</code>S = 72<code class='latex inline'>^\circ</code>, <code class='latex inline'>t=1.5</code>m, and <code class='latex inline'>s=1.8</code>m. Find <code class='latex inline'>\angle</code>T.</p>
<p>Solve the following triangles. Round lengths to the nearest tenth of a metre and angle measures to the nearest degree.</p><p>In <code class='latex inline'>\triangle WXY, w = 11 m</code>, <code class='latex inline'>x = 10m</code> and <code class='latex inline'>y =14 m</code>.</p>
<p><strong>a)</strong> A clock has a minute hand that is 20 cm long and an hour hand that is 12 cm long. Calculate the distance between the tips of the hands at</p> <ul> <li><em>i)</em> 2:00</li> <li><em>ii)</em> 10:00</li> </ul> <p><strong>b)</strong> Discuss your results for part a).</p>
<p><strong>a)</strong> Find <code class='latex inline'>x</code> to the nearest tenth of a centimetre.</p><p><strong>b)</strong> Find <code class='latex inline'>x</code> using a different method.</p><img src="/qimages/1621" />
<p>While cruising at a steady speed of <code class='latex inline'>400</code> km/h, you identify a storm cloud straight ahead 45 km away. To avoid turbulence, you start climbing at an angle of elevation of <code class='latex inline'>15</code><code class='latex inline'>^\circ</code>. If you maintain this speed and direction for <code class='latex inline'>6</code> min, how far will you be from the storm cloud? Round to the nearest kilometre.</p>
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