Textbook

10 Principles of Mathematics Nelson
Chapter

Chapter 2
Section

Analytic Geometry Chapter Review

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Solutions
25 Videos

On the design plan for a garden, a straight path runs from `(-25, 20)`

to `(40, 36)`

. A lamp is going to be placed at the midpoint of the path. Determine the coordinates for the lamp.

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0.00mins

Q1

`\triangle ABC`

has vertices at `A(-4, 4), B(-4, -2)`

, and `C(2, -2)`

.

a. Determine the equation of the median from `B`

to `AC`

.

b. Is the median for part a) also an altitude? Explain how you know.

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2.02mins

Q2

`\triangle LMN`

has vertices at `L(O, 4), M(-5, 2)`

, and `N(2, -2)`

. Determine the equation of the perpendicular bisector that passes through `MN`

.

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1.59mins

Q3

Which point is closer to the origin: `P(-24, 56)`

or `Q(35, -43)`

?

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1.01mins

Q4

A builder needs to connect a partially built house to a temporary power supply. On the plan, the coordinates of the house are `(20, 110)`

and the coordinates of the power supply are `(105, 82)`

. What is the least amount of cable needed?

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0.56mins

Q5

`\triangle QRS`

has vertices at Q(2, 6), R(—3, 1), and 5(6, 2). Determine the perimeter of the triangle.

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2.13mins

Q6

`\triangle XYZ`

as vertices at `X(1, 6)`

, `Y(-3, 2)`

, and `Z(9, 4)`

. Determine the length of the longest median in the triangle.

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2.33mins

Q7

a. Determine the equation of the circle that is centred at `(0, 0)`

and passes through point
`(-8, 15)`

.

b. Identify the coordinates of the intercepts and three other points on the circle.

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1.31mins

Q8

A circle has a diameter with endpoints `C(20, -21)`

and `D(-20, 21)`

. Determine the equation of the circle.

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1.51mins

Q9

Determine the equation of this circle.

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0.16mins

Q10

The point `(-2, k)`

lies on the circle `x^2 + y^2 = 20`

. Determine the values of `k`

. Show all the steps in your solution.

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0.50mins

Q11

`\triangle ABC`

has vertices as shown. Use analytic geometry to categorize `\triangle ABC`

. Show your work.

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1.03mins

Q12

A triangle has vertices at `A(1, 1), B(-2, -1)`

, and `C(3, -2)`

. Calculate the side lengths to determine whether the triangle is isosceles, equilateral, or scalene.

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1.12mins

Q13

Show that the quadrilateral with vertices at `J(-1, 1), K6, 4), L(8, 4)`

, and `M(4, 1)`

is a rhombus. What is the length of the side?

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2.13mins

Q14

Determine the type of quadrilateral described by the vertices `R(-3, 2), 5(- 1, 6), T(3, 5)`

, and `U(1, 1)`

. Show all the steps in your solution.

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2.52mins

Q15

A quadrilateral has vertices at `A(-3, 1), B(-5, -9), C(7, -1)`

, and `D(3, 3)`

. The mid segments of the quadrilateral form ?

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2.07mins

Q16

Which of the points `A(10, 10),B (-7, 3)`

, and `C(0, -14)`

lie on a circle with centre `O(5, -2)`

? What is the radius of this circle?

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2.26mins

Q17

A triangle has vertices at `P(-2, 7), Q(-4, 2)`

, and `R(6, -2)`

.

a. Categorize `\triangle PQR`

.

b. Show that the mid point of the hypotenuse is the same distance from each vertex. What is the distance?

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3.38mins

Q18

Show that points (6, 7) and (-9, 2) are the endpoints of a chord in a circle with centre (0, 0).

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3.03mins

Q19

Quadrilateral `JKZM`

has vertices as shown. Show that the diagonals of the quadrilateral bisect each other by showing they share the midpoint. What is the mid point?

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1.35mins

Q20a

`\triangle PQR`

has vertices at `P(0, -2), Q(4, 4)`

, and `R(-4, 5)`

. Use analytic geometry to determine
the coordinates of the orthocentre (the point where the altitudes intersect).

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5.56mins

Q21

`\triangle XYZ`

has vertices at `X(0, 1), Y(6, -1)`

, and `Z(3, 6)`

. Use analytic geometry to determine the coordinates of the centroid (the point where the medians intersect).

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1.41mins

Q22

A new lookout tower is going to be built so that it is the same distance from three ranger stations. If the stations are at `A(-90, 28), B(0, -35)`

, and `C(125, 20)`

on a grid, determine the coordinates of the point where the new tower should be built.

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5.47mins

Q23

Predict the type of quadrilateral that is formed by the points of intersection of the lines `3x+y-4=0`

, `4x-5y+30=0`

,
`4y = -3x -1`

, and `-4x + 5y +10 = 0`

. Give reasons for your prediction. Verify that your prediction is correct by solving this problem.

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2.11mins

Q24

A builder wants to run a temporary line from the main power line to a point near his site office. On the site plan, the site office is at `S(25, 18)`

and the main power line goes through points `T(1, 5)`

and `U(29, 12)`

. Each unit represents 1 m.

a. At what point should the builder connect to the main power line?

b. What length of cable will the builder need?

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3.17mins

Q25