Geometry

Ray C. Jurgensen,Richard G. Brown,John W. Jurgensen

Chapter 13

Coordinate Geometry - all with Video Answers

Educators

Section 1

The Distance Formula

Problem 1

Find the distance between the two points. If necessary, you may draw graphs but you shouldn't need to use the distance formula.
$$(-2,-3) \text { and }(-2,4)$$

Amrita Bhasin

Amrita Bhasin

Numerade Educator

Problem 2

Find the distance between the two points. If necessary, you may draw graphs but you shouldn't need to use the distance formula.
$$(3,3) \text { and }(-2,3)$$

Kami Dupree

Numerade Educator

Problem 3

Find the distance between the two points. If necessary, you may draw graphs but you shouldn't need to use the distance formula.
$$(3,-4) \text { and }(-1,-4)$$

Amrita Bhasin

Numerade Educator

Problem 4

Find the distance between the two points. If necessary, you may draw graphs but you shouldn't need to use the distance formula.
$$(0,0) \text { and }(3,4)$$

Kami Dupree

Numerade Educator

Problem 5

Use the distance formula to find the distance between the two points.
$$(-6,-2) \text { and }(-7,-5)$$

Amrita Bhasin

Numerade Educator

Problem 6

Use the distance formula to find the distance between the two points.
$$(3,2) \text { and }(5,-2)$$

Kami Dupree

Numerade Educator

Problem 7

Use the distance formula to find the distance between the two points.
$$(-8,6) \text { and }(0,0)$$

Amrita Bhasin

Numerade Educator

Problem 8

Use the distance formula to find the distance between the two points.
$$(12,-1) \text { and }(0,-6)$$

Kami Dupree

Numerade Educator

Problem 9

Find the distance between the points named. Use any method you choose.
$$(5,4) \text { and }(1,-2)$$

Amrita Bhasin

Numerade Educator

Problem 10

Find the distance between the points named. Use any method you choose.
$$(-2,-2) \text { and }(5,7)$$

Kami Dupree

Numerade Educator

Problem 11

Find the distance between the points named. Use any method you choose.
$$(-2,3) \text { and }(3,-2)$$

Amrita Bhasin

Numerade Educator

Problem 12

Find the distance between the points named. Use any method you choose.
$$(-4,-1) \text { and }(-4,3)$$

Kami Dupree

Numerade Educator

Problem 13

Given points $A, B,$ and $C .$ Find $A B, B C,$ and $A C .$ Are $A, B,$ and $C$ collinear? If so, which point lies between the other two?
$$A(0,3), B(-2,1), C(3,6)$$

Amrita Bhasin

Numerade Educator

Problem 14

Given points $A, B,$ and $C .$ Find $A B, B C,$ and $A C .$ Are $A, B,$ and $C$ collinear? If so, which point lies between the other two?
$$A(5,-5), B(0,5), C(2,1)$$

Kami Dupree

Numerade Educator

Problem 15

Given points $A, B,$ and $C .$ Find $A B, B C,$ and $A C .$ Are $A, B,$ and $C$ collinear? If so, which point lies between the other two?
$$A(-5,6), B(0,2), C(3,0)$$

Amrita Bhasin

Numerade Educator

Problem 16

Given points $A, B,$ and $C .$ Find $A B, B C,$ and $A C .$ Are $A, B,$ and $C$ collinear? If so, which point lies between the other two?
$$A(3,4), B(-3,0), C(-1,1)$$

Kami Dupree

Numerade Educator

Problem 17

Find the center and the radius of each circle.
$$(x+3)^{2}+y^{2}=49$$

Amrita Bhasin

Numerade Educator

Problem 18

Find the center and the radius of each circle.
$$(x+7)^{2}+(y-8)^{2}=\frac{36}{25}$$

Kami Dupree

Numerade Educator

Problem 19

Find the center and the radius of each circle.
$$(x-j)^{2}+(y+14)^{2}=17$$

Amrita Bhasin

Numerade Educator

Problem 20

Find the center and the radius of each circle.
$$(x+a)^{2}+(y-b)^{2}=c^{2}$$

Kami Dupree

Numerade Educator

Problem 21

Write an equation of the circle that has the given center and radius.
$$C(3,0) ; r=8$$

Amrita Bhasin

Numerade Educator

Problem 22

Write an equation of the circle that has the given center and radius.
$$C(0,0) ; r=6$$

Kami Dupree

Numerade Educator

Problem 23

Write an equation of the circle that has the given center and radius.
$$C(-4,-7) ; r=5$$

Amrita Bhasin

Numerade Educator

Problem 24

Write an equation of the circle that has the given center and radius.
$$C(-2,5) ; r=\frac{1}{3}$$

Kami Dupree

Numerade Educator

Problem 25

Sketch the graph of $(x-3)^{2}+(y+4)^{2}=36$.

Amrita Bhasin

Numerade Educator

Problem 26

Sketch the graph of $(x-2)^{2}+(y-5)^{2} \leq 9$.

Kami Dupree

Numerade Educator

Problem 27

In Exercises 27-32 find and then compare lengths of segments.
Show that the triangle with vertices $A(-3,4), M(3,1),$ and $Y(0,-2)$ is isosceles.

Amrita Bhasin

Numerade Educator

Problem 28

In Exercises 27-32 find and then compare lengths of segments.
Quadrilateral $T A U L$ has vertices $T(4,6), A(6,-4), U(-4,-2),$ and $L(-2,4) .$ Show that the diagonals are congruent.

Kami Dupree

Numerade Educator

Problem 29

In Exercises 27-32 find and then compare lengths of segments.
Triangles $J A N$ and $R F K$ have vertices $J(-2,-2), A(4,-2), N(2,2),$ $R(8,1), F(8,4),$ and $K(6,3)$ . Show that $\triangle J A N$ is similar to $\triangle R F K$ .

Amrita Bhasin

Numerade Educator

Problem 30

In Exercises 27-32 find and then compare lengths of segments.
The vertices of $\triangle K A T$ and $\triangle I E S$ are $K(3,-1), A(2,6), T(5,1),$ $I(-4,1), E(-3,-6),$ and $S(-6,-1),$ What word best describes the relationship between $\triangle K A T$ and $\triangle I E S ?$

Kami Dupree

Numerade Educator

Problem 31

In Exercises 27-32 find and then compare lengths of segments.
Find the area of the rectangle with vertices $B(8,0), T(2,-9),$ $R(-1,-7),$ and $C(5,2) .$

Amrita Bhasin

Numerade Educator

Problem 32

In Exercises 27-32 find and then compare lengths of segments.
Show that the triangle with vertices $D(0,0), E(3,1),$ and $F(-2,6)$ is a right triangle, then find the area of the triangle.

Kami Dupree

Numerade Educator

Problem 33

There are twelve points, each with integer coordinates, that are 10 units from the origin. List the points. (Hint: Recall the $6,8,10$ right triangle.)

Amrita Bhasin

Numerade Educator

Problem 34

a. List twelve points, each with integer coordinates, that are 5 units from $(-8,1)$ .
b. Find an equation of the circle containing these points.

Kami Dupree

Numerade Educator

Problem 35

In Exercises 35-38 find an equation of the circle described and sketch the graph.
The circle has center $(0,6)$ and passes through point $(6,14) .$

Amrita Bhasin

Numerade Educator

Problem 36

In Exercises 35-38 find an equation of the circle described and sketch the graph.
The circle has center $(-2,-4)$ and passes through point $(3,8)$ .

Kami Dupree

Numerade Educator

Problem 37

In Exercises 35-38 find an equation of the circle described and sketch the graph.
The circle has diameter $\overline{R S}$ where $R$ is $(-3,2)$ and $S$ is $(3,2)$ .

Amrita Bhasin

Numerade Educator

Problem 38

In Exercises 35-38 find an equation of the circle described and sketch the graph.
The circle has center $(p, q)$ and is tangent to the $x$ -axis.

Kami Dupree

Numerade Educator

Problem 39

a. Find the radii of the circles
$$x^{2}+y^{2}=25 \text { and }(x-9)^{2}+(y-12)^{2}=100$$
b. Find the distance between the centers of the circles.
c. Explain why the circles must be externally tangent.
d. Sketch the graphs of the circles.

Kami Dupree

Numerade Educator

Problem 40

a. Find the radii of the circles
$$x^{2}+y^{2}=2 \text { and }(x-3)^{2}+(y-3)^{2}=32$$
b. Find the distance between the centers of the circles.
c. Explain why the circles must be internally tangent.
d. Sketch the graphs of the circles.

Kami Dupree

Numerade Educator

Problem 41

Discover and prove something about the quadrilateral with vertices $R(-1,-6), A(1,-3), Y(11,1),$ and $J(9,-2)$

Amrita Bhasin

Numerade Educator

Problem 42

Discover and prove two things about the triangle with vertices $K(-3,4)$, $M(3,1),$ and $J(-6,-2) .$

Kami Dupree

Numerade Educator

Problem 43

It is known that $\triangle G H M$ is isosceles. $G$ is point $(-2,-3), H$ is point $(-2,7),$ and the $x$ -coordinate of $M$ is $4 .$ Find all five possible values for the $y$ -coordinate of $M .$

Amrita Bhasin

Numerade Educator

Problem 44

Find the coordinates of the point that is equidistant from $(-2,5),(8,5),$ and $(6,7) .$

Kami Dupree

Numerade Educator

Problem 45

Find the center and the radius of the circle $x^{2}+4 x+y^{2}-8 y=16$ . (Hint: Express the given equation in the form
$$(x-a)^{2}+(y-b)^{2}=r^{2} . )$$

Amrita Bhasin

Numerade Educator