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Solutions
164 Videos

Translation each sentence into an equation.

Tell how you are assigning the two variables.

a) The perimeter of a basketball court is `40`

m.

b) The average of two numbers is `15`

.

c) The value of the quarters and loonies in a vending machine is `\$37`

.

d) The total receipts from adult tickets at `\$20`

each and student tickets at `\$12`

each was `\$9250`

.

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Q1

Find the intersection point.

```
\displaystyle
x -y =4
```

```
\displaystyle
3x + 2y =7
```

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Q2a

Find the intersection point.

```
\displaystyle
y= \frac{1}{2}x -4
```

```
\displaystyle
y = 2x- 1
```

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Q2b

Find the intersection point.

```
\displaystyle
x + y - 4= 0
```

```
\displaystyle
5x -y -8=0
```

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Q2c

Solve each linear system using the method of substitution.

```
\displaystyle
x + 4y =6
```

```
\displaystyle
2x -3y =1
```

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Q3a

Solve each linear system using the method of substitution.

```
\displaystyle
y = 6-3x
```

```
\displaystyle
y = 2x +1
```

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Q3b

Solve each linear system using the method of substitution.

```
\displaystyle
5x - y = 4
```

```
\displaystyle
3x + y =4
```

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Q3c

Solve by elimination. Check each solution.

```
\displaystyle
x + y = 55
```

```
\displaystyle
2x - y = -4
```

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Q4a

Solve by elimination. Check each solution.

```
\displaystyle
2a +b = 5
```

```
\displaystyle
a - 2b = 10
```

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Q4b

Solve by elimination. Check each solution.

```
\displaystyle
4k + 3h =12
```

```
\displaystyle
4k -h =4
```

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Q4c

Solve by elimination. Check each solution.

```
\displaystyle
5a -2b = 5
```

```
\displaystyle
3a +2b = 19
```

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Q4d

Explain why the following linear system has no solution.

```
\displaystyle
y - 2x = 1
```

```
\displaystyle
y = 2x + 3
```

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Q5

Use a graphing calculator to find the point of intersection of each pair of lines.

```
\displaystyle
y = x -5
```

```
\displaystyle
x + 2y =10
```

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Q6a

Use a graphing calculator to find the point of intersection of each pair of lines.

```
\displaystyle
2x +5y + 20 = 0
```

```
\displaystyle
5x - 3y + 15 = 0
```

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Q6b

Use a graphing calculator to find the point of intersection of each pair of lines.

```
\displaystyle
y = 7x
```

```
\displaystyle
3y =5x -2
```

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Q6c

Find the values of `a`

and `b`

in the diagram shown.

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Q7

A boat took 5 h to travel 60 km up a river, against the current. The return trip took 3 h.

Find the speed of the boat in still water and the speed of the current.

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Q8

What volumes of 60% hydrochloric acid solution and 30% hydrochloric acid solution must be mixed to make 125 mL of 36% hydrochloric acid solution?

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Q9

Solve the linear system.

```
\displaystyle
\frac{x-2}{3} + \frac{y + 1}{5} =2
```

```
\displaystyle
\frac{x + 2}{7} - \frac{y + 5}{3} = -2
```

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Q10

Find the coordinates of the midpoint and the length of each line segment.

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Q11

For `\triangle`

, find an equation for the median from vertex `J`

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Q12a

For `\triangle`

, find an equation for the median from vertex `K`

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Q12b

For `\triangle`

, the right bisector of side `JL`

.

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Q12c

On a street map, the coordinates of the two fire stations in a town are A(10, 63) and B(87, 30). A neighbor reports smoke coming from the kitchen of a house at C(41, 18).

a) Which fire station is closer to this house?

b) Describe how to use geometry software in answer part a).

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Q13

Use analytic geometry to classify the quadrilateral with vertices `D(10, 0), E(2, 4). F(-8, -6)`

, and `G(6, -8)`

. Explain your reasoning and show all your work.

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Q14

a) Draw the triangle with vertices `J(2, 10), K((6, -6)`

, and `L(14, 6)`

.

b) Calculate the coordinates of the midpoint, M, of side JK and the coordinates of the midpoint, `N`

, of side `JL`

.

c) Show that `MN`

is half the length of `KL`

.

d) Show that `MN`

is parallel to `KL`

.

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Q15ab

Triangle with vertices J(2, 10], K((6, -6), and L(14, 6).

a) Show that MN is half the length of KL.

b) Show that MN is parallel to KL.

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Q15cd

Does the point `P(-3, -2)`

lie on the right bisector of the line segment with endpoints `Q(-2, 5)`

and `R(4, 1)`

? Justify your answer.

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Q16

Determine the type of quadrilateral that has vertices at `A(4, 6), B(-4, -2)`

, `C(2, -5)`

, and `D(11, 4)`

.

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Q17

On a site plan for a new house, water main runs along the edge of the property straight from point `W(10, 34)`

to point `M(2,2)`

. The water service will enter the house at point `H(24, 22)`

. The grid intervals on the plan represent 0.5 m.

At what point on the water main should the connector to take water to the house be located?

Explain your reasoning.

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Q18a

On a site plan for a new house, water main runs along the edge of the property straight from point `W(10, 34)`

to point `M(2,2)`

. The water service will enter the house at point `H(24, 22)`

. The grid intervals on the plan represent 0.5 m.

What length of pipe will this connection require?

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Q18b

Find an equation for the circle centred at the origin and passing through the point

a) `J(0, 7)`

b) `K(5, 6)`

c) `(8, \sqrt{3})`

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Q19

Find the diameter and the area of the circle defined by `x^2 + y^2 = 64`

.

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Q20

You are designing a chute for loading grain into rail cars. The cars have a round hatch 60 cm in diameter. The chute will have a square cross section. Find the side length of the largest chute that will fit into the hatch. Round your answer to the nearest centimetre.

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Q21

a) What is the centroid of a triangle?

b) Describe how to use analytic geometry to find the coordinates of the centroid of a triangle, given the coordinates of the vertices.

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Q22

a) Show that each median of a triangle bisects the area of the triangle.

b) Outline how to use geometry software to answer part a).

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Q23

Verify that `\triangle ABC`

is isosceles.

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Q24

Verify that `\triangle DEF`

is a right triangle using slopes.

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Q25a

Verify that `\triangle DEF`

is a right triangle using the Pythagorean Theorem.

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Q25b

a) Draw the triangle with vertices `J(-4, -6), K(2, -4)`

, and `L(8, 10)`

.

b) Verify that the triangle formed by joining the midpoints of the sides of `\triangle JKL`

is similar to `\triangle JKL`

.

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Q26

a) Find an equation for each of the right bisectors of the sides of `\triangle JKL`

.

b) Find the Circumcentre of `\triangle PQR`

, the point of intersection of the right
bisectors of the three sides.

c) Show that the circumcentre of `\triangle PQR`

is equidistant from its vertices.

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Q27

List the types of quadrilaterals that have diagonals that

a) are equal in length

b) bisect each other

c) meet at right angles

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Q28

a) Show that a line segment that joins the midpoints of opposite sides of a parallelogram has the same length as the other two sides of the parallelogram.

b) Outline how to use geometry software to answer part a).

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Q29

a) Verify that the quadrilateral with vertices `A(2, 7), B(-2, 2), C(4, -2)`

, and `D(8, 3)`

is a rectangle.

b) Verify that the diagonals of `ABCD`

are equal in length.

c) Verify that the diagonals bisect each other.

d) Verify that the diagonals are not perpendicular.

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Q30

Determine the type of quadrilateral created by the points of intersection of the lines

```
\displaystyle
x - y + 3= 0
```

, ```
\displaystyle
y = x- 2
```

, ```
\displaystyle
y = - \frac{1}{2}x + 5
```

and ```
\displaystyle
x + 2y + 12 = 0
```

.

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Q31

a) Verify that the points `P(2, 5), Q(6, -1)`

, and `R(7, 4)`

all lie on a circle centred at `C(4, 2)`

.

b) Verify that the centre of the circle lies on the right bisector of Chord `PQ`

.

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Q32

Outline how to use geometry software to find the centre of a circle given the coordinates of three points on the circle.

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Q33

Use finite differences to determine whether each relation is linear, quadratic, or neither.

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Q34a

Use finite differences to determine whether each relation is linear, quadratic, or neither.

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Q34b

Use finite differences to determine whether each relation is linear, quadratic, or neither.

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Q34c

A basketball shot is taken from a horizontal distance of `5`

m from the hoop. The height of the ball can be modelled by the relation `h = -7.3t^2 + 8.25t + 2.1`

, where `h`

is the height, in metres, and `t`

is the time, in seconds, since the ball was released.

a) From what height was the ball released?

b) What was the maximum height reached by the ball?

c) If the ball reached the hoop in `1`

s, what was the height of the hoop?

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Q35

Sketch the parabola. Label the coordinates of the vertex and the equation of the axis of symmetry.

```
\displaystyle
y = x^2 -3
```

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Q36a

Sketch the parabola. Label the coordinates of the vertex and the equation of the axis of symmetry.

```
\displaystyle
y = (x+ 1)^2 -1
```

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Q36b

Sketch the parabola. Label the coordinates of the vertex and the equation of the axis of symmetry.

```
\displaystyle
y = -2(x -4)^2 -3
```

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Q36c

Sketch the parabola. Label the coordinates of the vertex and the equation of the axis of symmetry.

```
\displaystyle
y = 0.5(x+6)^2 +3
```

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Q36d

The height of lava ejected from the Stromboli volcano can be modelled by the relation `h = -5t(t - 11)`

. where `h`

is the height, in metres. of the lava above the crater and `t`

is the time,. in seconds, since it was ejected.

a) Graph the relation.

b) Find the maximum height reached by the lava, to the nearest metre.

c) How long does the lava take to reach the maximum height?

d) Is the length of time that the lava is in the air twice the answer from part (c)? Explain why or why not.

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Q37

Determine an equation to represent the parabola.

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Q38a

Determine an equation to represent the parabola.

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Q38b

The vertex of a parabola `(-2, -4)`

. One x-intercept is 7. What is the other x-intercept?

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Q39

Evaluate.

`8^o`

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Q40a

Evaluate.

`3^{-1}`

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Q40b

Evaluate.

`(-2)^{-5}`

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Q40c

Evaluate.

`(\frac{2}{5})^{-2}`

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Q40d

Evaluate.

`(-18)^{0}`

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Q40e

Evaluate.

`(\frac{4}{3})^{-1}`

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Q40f

A piece of wood burns completely in `1`

s
at 600°C. The time it takes for the wood to burn is doubled for every 10°C drop in A temperature and halved for every 10`^oC`

increase in temperature. Determine how long the piece of wood would take to burn‘ at each temperature.

a) `500^oC`

b) `650^oC`

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Q41

Expand and simplify.

```
\displaystyle
3(x-4)+5(x + 6)
```

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Q42a

Expand and simplify.

```
\displaystyle
6(a+ 3)-2(a-5)
```

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Q42b

Expand and simplify.

```
\displaystyle
4(1 -3k)-(2 -4k)
```

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Q42c

Expand and simplify.

```
\displaystyle
2t(3t -4)+t(2t +5)
```

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Q42d

Expand and simplify.

```
\displaystyle
\frac{1}{2}(6 +3p) - \frac{3}{4}(8 - 10p)
```

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Q42e

Expand and simplify.

```
\displaystyle
3y(y^2 - y -1)-y(2y^2 - 3y + 4)
```

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Q42f

a) Write an algebraic expression to represent the surface area of the rectangular prism. Expand and simplify.

b) If `x`

represents 5 cm, what is the surface area?

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Q43

Expand and simplify.

```
\displaystyle
(x+ 4)^2
```

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Q44a

Expand and simplify.

```
\displaystyle
(x- 4)(x + 4)
```

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Q44b

Expand and simplify.

```
\displaystyle
(x - 5)^2
```

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Q44c

Expand and simplify.

```
\displaystyle
(3t-1)(3t + 1)
```

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Q44d

Expand and simplify.

```
\displaystyle
(4a + 3b)(5a - 3b)
```

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Q44e

Expand and simplify.

```
\displaystyle
2(3m + 1)^2
```

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Q44f

Expand and simplify.

```
\displaystyle
(m -3)(m +3) + (m -4)^2
```

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Q45a

Expand and simplify.

```
\displaystyle
(2t + 1)^2 + 2(3t-1)(3t + 1)
```

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Q45b

Expand and simplify.

```
\displaystyle
2(2x + 2y)(3x - 2y) - 3(3x - y)^2
```

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Q45c

Expand and simplify.

```
\displaystyle
(2y -1)^2 + (y + 2)^2 - (1-2y)(1+ 2y)
```

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Q45d

Expand and simplify.

```
\displaystyle
(4m + n)^2 + 2(m -2n)^2
```

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Q45e

Expand and simplify.

```
\displaystyle
5(2t - 5z)^2 + 3(4t-3z)(4t + 3z)
```

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Q45f

Factor.

a) ```
\displaystyle
5k 35
```

b) `4h^2 -20h`

c) `2xy - 8xy^2`

d) `x^2 - 25`

e) `1 - 49 m^2`

f) `4a^2 - 16b^2`

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Q46

The area of the rectangle is given by the expression `n^2 -5n + 6`

.

a) Find expressions for the length and the width.

b) If `n`

represents 8 cm, determine the perimeter and the area of the rectangle.

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Q47

Factor.

```
\displaystyle
x^2 -x - 12
```

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Q48a

Factor.

```
\displaystyle
x^2 + 2x - 18
```

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Q48b

Factor.

```
\displaystyle
m^2 + 11m + 24
```

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Q48c

Factor.

```
\displaystyle
t^2 - 8t + 15
```

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Q48d

Factor.

```
\displaystyle
x^2+ 3x + 4
```

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Q48e

Factor.

```
\displaystyle
n^2 - 13n + 40
```

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Q48f

Factor.

```
\displaystyle
w^2 - w- 30
```

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Q48g

Factor.

```
\displaystyle
14 + 5m - m^2
```

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Q48h

Factor, if possible.

```
\displaystyle
x^2 + 10 x + 25
```

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Q49a

Factor, if possible.

```
\displaystyle
y^2 -12 y + 36
```

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Q49b

Factor, if possible.

```
\displaystyle
m^2 + 6m + 16
```

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Q49c

Factor, if possible.

```
\displaystyle
4x^2 + 12x + 9
```

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Q49d

Factor, if possible.

```
\displaystyle
25r^2 - 20rs + 4s^2
```

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Q49e

Factor, if possible.

```
\displaystyle
5x^2 -20xy + 2y^2
```

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Q49f

Write two different trinomials that have `x + 2`

as a factor.

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Q50

Determine all values of `p`

so that each trinomial is a perfect square.

```
\displaystyle
x^2 + px + 16
```

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Q51a

Determine all values of `p`

so that each trinomial is a perfect square.

```
\displaystyle
4x^2 - 12x + p
```

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Q51b

Determine all values of `p`

so that each trinomial is a perfect square.

```
\displaystyle
px^2 + 40x + 16
```

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Q51c

If `m`

and `n`

are integers, find values of `m`

and `n`

such that `m^2 -n^2 = 21`

.

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Q52

Rewrite the relation in the form `y = a(x - h)^2 + k`

by completing the square.

Graph the relation and give the vertex and the equation of the axis of symmetry.

```
\displaystyle
y = x^2 +4x + 1
```

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Q53a

Rewrite the relation in the form `y = a(x - h)^2 + k`

by completing the square.

Graph the relation and give the vertex and the equation of the axis of symmetry.

```
\displaystyle
y = -x^2 - 6x - 5
```

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Q53b

Rewrite the relation in the form `y = a(x - h)^2 + k`

by completing the square.

Graph the relation and give the vertex and the equation of the axis of symmetry.

```
\displaystyle
y = 3 - 4x -x^2
```

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Q53c

Find the x-intercepts of the parabola.

```
\displaystyle
y = x^2 + 2x -3
```

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Q55a

Find the x-intercepts of the parabola.

```
\displaystyle
y =x^2 + 6x + 5
```

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Q55b

Find the x-intercepts of the parabola.

```
\displaystyle
y = x^2 -4x + 4
```

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Q55c

Find the x-intercepts of the parabola.

```
\displaystyle
y = 4x^2 -12x + 9
```

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Q55d

Solve by factoring. Check your solutions.

```
\displaystyle
x^2 + 3x - 28 = 0
```

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Q56a

Solve by factoring. Check your solutions.

```
\displaystyle
m^2 + 7m + 10 = 0
```

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Q56b

Solve by factoring. Check your solutions.

```
\displaystyle
2n^2 = 27 - 15n
```

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Q56c

Solve by factoring. Check your solutions.

```
\displaystyle
3k(k - 4) + k + 4(k + 1) = 0
```

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Q56d

Write a quadratic equation with the given roots.

```
\displaystyle
5, -2
```

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Q57a

Write a quadratic equation with the given roots.

```
\displaystyle
- \frac{3}{4}, - \frac{2}{3}
```

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Q57b

Find the value of `k`

so that the parabola has only one x-intercept.

```
\displaystyle
kx^2 - 5x + 2= 0
```

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Q58a

Find the value of `k`

so that the parabola has only one x-intercept.

```
\displaystyle
x^2 + kx + 9 = 0
```

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Q58b

Find the value of `k`

so that the parabola has only one x-intercept.

```
\displaystyle
25x^2 + 20x + k = 0
```

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Q58c

The area of the rectangle is `36 cm^2`

. What are its dimensions?

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Q59

Use the quadratic formula to solve the equation. Express your answers as exact roots.

```
\displaystyle
x^2 - x -4= 0
```

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Q60a

Use the quadratic formula to solve the equation. Express your answers as exact roots.

```
\displaystyle
7k^2 -2k - 2 = 0
```

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Q60b

Use the quadratic formula to solve the equation. Express your answers as exact roots.

```
\displaystyle
2x^2 = 3 -8x
```

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Q60c

Use the quadratic formula to solve the equation. Express your answers as exact roots.

```
\displaystyle
4h = 5- 4h^2
```

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Q60d

Use the quadratic formula to solve the equation. Express your answers as exact roots.

```
\displaystyle
0 = -3a^2 +4a + 1
```

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Q60e

A store sells 90 ski jackets for $200 each. A survey indicates that for each $10 decrease in price, five more jackets will be sold.

a) Find the number of jackets sold and the selling price when the total revenue is $17 600.

b) What is the lowest price that will give revenues of at least $15 600? How many jackets would be sold at this price?

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Q61

The hypotenuse of a right triangle measures 20 cm. The sum of the lengths of the legs is 28 cm. Find the length of each leg of the triangle.

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Q62

A rectangular swimming pool measuring 10m by 4m is surrounded by a deck of uniform width. The total area of the pool and deck is 135 `m^2`

. What is the width of the deck?

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Q63

Identify the two similar triangles and explain why they are similar.

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Q64

At noon, the shadow of a tree is 4.3 m long. At the same time, the shadow of a metre stick is 0.2 m long. What is the height of the tree?

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Q65

Find the measure of `\angle A`

, to the nearest degree.

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Q66a

Find the measure of `\angle A`

, to the nearest degree.

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Q66b

A road sign shows that a hill has a grade of 8%. What is the angle of inclination of the hill, to the nearest tenth of a degree?

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Q67

Find x, to the nearest tenth of a unit.

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Q68a

Find x, to the nearest tenth of a unit.

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Q68b

Solve the triangle. Round lengths to the nearest unit and angle measures to the nearest degree.

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Q69a

Solve the triangle. Round lengths to the nearest unit and angle measures to the nearest degree.

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Q69b

Solve the triangle. Round lengths to the nearest unit and angle measures to the nearest degree.

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Q69c

Solve the triangle. Round lengths to the nearest unit and angle measures to the nearest degree.

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Q69d

In `\triangle XYZ`

, `\angle X =`

90°, `XY = 3.5 cm`

, and `YZ = 4.8 cm`

. Solve `\triangle XYZ`

. Round lengths to the nearest tenth of a metre and angle measures to the newest degree.

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Q70

In `\triangle ABC`

, `\angle C = 90^o`

. If `\sin A = 0.5`

, what is the measure of `\angle B`

?

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Q71

A coast guard boat is 14.8 km east of a lighthouse. A disabled yacht is 7.5 km south of the lighthouse.

a) How far is the coast guard boat from the yacht, to the nearest tenth of a kilometre?

b) At what angle south of due west, to the nearest tenth of a degree, should the coast guard boat travel to reach the yacht?

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Q72

The Confederation Bridge joins New Brunswick and Prince Edward Island. From a boat in the Northumberland Strait, the angle of elevation of the highest point on the bridge is `26.6^o`

. When the boat is 100 m closer to the bridge, the angle of elevation is `71.7^o`

. What is the height of the bridge, to the nearest tenth of a metre?

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Q73

Find the length of x, to the nearest centimetre.

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Q74

Find the measure of `\angle A`

. to the nearest degrees.

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Q75

In `\triangle ABC`

, `\angle B = 38.2^o`

, `\angle C = 65.6^o`

, and `b = 54 cm`

. Find `c`

, to the nearest tenth of a centimetre.

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Q76

Solve the triangle.

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Q77a

Solve the triangle.

In `\triangle JKL, \angle L = 32^o`

, `j = 20.5 cm`

, and `\angle K = 75^o`

.

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Q77b

The longer diagonal of a parallelogram measures 8.5 cm. This diagonal makes angles of 43° and 32° with the sides of the parallelogram, as shown. Find the length of each side of the parallelogram, to the nearest tenth of a centimetre.

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Q78

Find the unknown side length, to the nearest tenth of a metre.

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Q79

Solve `\triangle KLM`

. Round answers to the nearest tenth of a degree.

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Q80

Determine the area of `\triangle RST`

, to the nearest square metre.

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Q81

Determine the perimeter of isosceles `\triangle ABC`

, to the nearest tenth of a centimetre.

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Q82

Find the length `x`

, to the nearest tenth of a metre.

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Q83a

Find the length `x`

, to the nearest tenth of a metre.

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Q83b

Two boats left the harbour at the same time. One travelled at 10 km/h on a bearing of `N47^oE`

.

The other travelled at 8 km/h on a bearing of `N79^oE`

.

How far apart were the boats after 45 min? Round the distance to the nearest tenth of a kilometre.

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Q84

Solve the following triangles. Round lengths to the nearest tenth of a metre and angle measures to the nearest degree.

In `\triangle DEF, \angle D = 58^o, \angle E = 53^o`

, and `d = 8 cm`

.

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Q85a

Solve the following triangles. Round lengths to the nearest tenth of a metre and angle measures to the nearest degree.

In `\triangle RST, \angle R = 73^o, r = 8m`

and `t = 6 m`

.

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Q85b

Solve the following triangles. Round lengths to the nearest tenth of a metre and angle measures to the nearest degree.

In `\triangle ABC, \angle A =68^o`

, `b= 5 cm`

and `c =7 cm`

.

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Q85c

In `\triangle WXY, w = 11 m`

, `x = 10m`

and `y =14 m`

.

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Q85d

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.

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Q86