Chapter

Chapter 2
Section

Chapter Review Analytic Geometry

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Solutions
35 Videos

Find the midpoint of the line segment.

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Q1a

Find the midpoint of the line segment.

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Q1b

Find the midpoint of two given points below.

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Q1c

Find the midpoint of two given points below.

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Q1d

Determine the midpoint of the line segment with endpoints `J(3, -5)`

and `K(-5, -6)`

.

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Q2a

Determine the midpoint of the line segment with endpoints `L(4, 8)`

and `N(4, -2)`

.

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Q2b

a) Draw the triangle with vertices `P(-2, 5)`

, `Q(6, 5)`

, and `R(2, -7)`

.

b) Determine the midpoint of each side of the triangle algebraically.

c) Join the midpoints to form a smaller triangle. Compare this triangle to the original triangle.

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Q3

a) Draw the triangle with vertices `T(-8, 6), U(2, 10)`

, and `W(4, -4)`

.

b) Draw the median from vertex `U`

. Then, find an equation for this median.

c) Draw the altitude from vertex `T`

. Then, find an equation for this altitude.

d) Draw the right bisector of `TU`

. Then, find an equation for this right bisector.

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Q4

The midpoints of the sides of `\triangle ABC`

are `D(4, 1), E(-2, 3)`

, and `F(1, -4)`

.

a) Plot the midpoints. Use this plot to estimate the coordinates of the vertices of `\triangle ABC`

.

b) Use analytic geometry to calculate the coordinates of the vertices of `\triangle ABC`

.

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Q5

Find the length of the line segment `\overline{AB}`

where `A(3, 17), B(8, 5)`

.

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Q6a

Find the length of each line segment.

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Q6b

Find the length of each line segment.

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Q6c

Find the length of each line segment.

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Q6d

Determine the length of the line segment defined by each pair of points.

`J(4, 8)`

and `K(4, -2)`

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Q7a

Determine the length of the line segment defined by each pair of points.

`M(-3, -12)`

and `N(-15, -7)`

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Q7b

Determine the length of the line segment defined by each pair of points.

`P(-3, -2)`

and `Q(5, 6)`

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Q7c

Determine the length of the line segment defined by each pair of points.

`R(-1, 5)`

and `S(4, -1)`

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Q7d

Determine the length of the line segment defined by each pair of points.

`T(-2, 4)`

and `U(7, 4)`

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Q7e

Determine the length of the line segment defined by each pair of points.

`V(3, -5)`

and `W(-5, -6)`

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Q7f

Determine the length of the median from vertex `A`

of `\triangle ABC`

.

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Q8a

a) Draw the triangle with vertices `D(5, 25), E(210,1)`

, and `F(3, 210)`

.

b) Use analytic geometry to Classify `\triangle DEF`

.

c) Determine the area of `\triangle DEF`

.

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Q9

Show that this triangle is isosceles.

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Q10

A triangle has vertices `D(-2, 7), E(-4, 2)`

, and `F(6, -2)`

.

a) Show algebraically that this triangle is a right triangle.

b) Find the midpoint of the hypotenuse.

c) Show that this midpoint is equidistant from each of the vertices.

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Q11

A map shows a main gas pipeline running straight from `A(45, 60)`

to `B(65, 40)`

.

a) How long is the section of pipeline from A to B if each unit on the map grid represents 1 km?

b) A branch pipeline runs perpendicular to the main pipeline and meets it at a point halfway between A and B. Find the coordinates of this point.

c) Is the point `C(63, 54)`

on the branch pipeline? Explain your reasoning.

d) What is the shortest route for connecting point C to the main pipeline? Explain.

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Q12

Find the shortest distance from the origin to the line defined by `y = 3x - 10`

.

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Q13

Determine the equation for the circle.

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Q14a

Determine the equation for the circle.

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Q14b

Determine the equation for the circle.

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Q14c

Find an equation for the circle that is centred on the origin and has a radius of `4.5`

.

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Q15a

Find an equation for the circle that is centred on the origin and ha a diameter of `14`

.

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Q15b

Find an equation for the circle that is centred on the origin and has a radius of `\sqrt{12}`

.

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Q15c

Find an equation for the circle that is centred on the origin and passes through the point `(4, 7)`

.

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Q15d

a) Determine whether the point `A(-2, -6)`

lies on the circle defined by `x^2 + y^2 = 40`

.

b) Find an equation for the radius from the origin `O`

to point `A`

.

c) Find an equation for the line that passes through `A`

and is perpendicular to `OA`

.

d) Use a graph to check your answers to parts a), b), and c).

e) Explain why `A`

is the only point on the line that also lies on the circle.

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Q16

a) Show that the line segment joining `A(-3, 1)`

and `B(1, 3)`

is a chord of the circle defined by `x^2 + y^2 = 10`

b) Determine an equation for the right bisector of the chord `AB`

.

c) Show that the line in part b) passes through the centre of the circle.

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Q17

A communication tower can send and receive signals from cell phones up to `20`

km away. A cell phone user is `15`

km east and `13`

km south of the tower. Is this user able to receive a signal from the tower?

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Q18