# Geometry

## Educators  ### 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 B.

### 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 D.

### 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 B.

### 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 D.

### Problem 5

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

### Problem 6

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

### Problem 7

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

### Problem 8

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

### Problem 9

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

### Problem 10

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

### Problem 11

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

### Problem 12

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

### 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 B.

### 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 D.

### 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 B.

### 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 D.

### Problem 17

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

### Problem 18

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

### Problem 19

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

### Problem 20

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

### Problem 21

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

### Problem 22

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

### Problem 23

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

### Problem 24

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

### Problem 25

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

### Problem 26

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

### 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 B.

### 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 D.

### 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 B.

### 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 D.

### 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 B.

### 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 D.

### 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 B.

### 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 D.

### 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 B.

### 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 D.

### 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 B.

### 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 D.

### 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 D.

### 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 D.

### 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 B.

### Problem 42

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

### 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 B.

### Problem 44

Find the coordinates of the point that is equidistant from $(-2,5),(8,5),$ and $(6,7) .$ Kami D.
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
$$(x-a)^{2}+(y-b)^{2}=r^{2} . )$$ 