Topics
12. Circles
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Geometry

12.03 Circles in the coordinate plane

Lesson
Questions
Adaptive
Worksheet
What do you remember?
1

Find the radius of the circles given the following information:

a

Center: \left(0,0\right)

Point on circle: \left(-5,0\right)

b

Center: \left(-3,0\right)

Point on circle: \left(-3,-1\right)

2

What is the diameter of the circle?

A
7
B
4
C
8
D
5
-5
-4
-3
-2
-1
1
2
3
4
5
x
-1
1
2
3
4
5
6
7
8
9
y
3

Find the center and radius of each circle:

a
-6
-5
-4
-3
-2
-1
1
x
-6
-5
-4
-3
-2
-1
1
y
b
-4
-3
-2
-1
1
2
3
4
x
-3
-2
-1
1
2
3
4
5
y
c
-3
-2
-1
1
2
3
x
-4
-3
-2
-1
1
2
y
d
-10
-8
-6
-4
-2
2
x
-2
2
4
6
8
10
y
4

Using the graph of the circle:

a

What are the coordinates of the center?

b

What is the length of the diameter?

-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
y
Let's practice
5

Match the given coordinates of the endpoints of diameters with the correct center coordinates of the circles.

a

Endpoints \left(3,6\right) and \left(9,6 \right)

b

Endpoints \left(-2,4\right) and \left(4,4 \right)

c

Endpoints \left(0,-1\right) and \left(6,-1 \right)

d

Endpoints \left(-5,2\right) and \left(1,2 \right)

e

Endpoints \left(2,-3\right) and \left(8,-3 \right)

i
Center \left(-2,2 \right)
ii
Center \left(3,-1 \right)
iii
Center \left(5, -3 \right)
iv
Center \left(6,6 \right)
v
Center \left(1,4 \right)
6

Find the centers of the circles with the following diameter endpoints:

a

\left(0,0\right) and \left(0,4\right)

b

\left(-4, 2\right) and \left(6, 4\right)

c

\left(\dfrac{16}{5}, -\dfrac{8}{5}\right) and \left(\dfrac{14}{5},-\dfrac{2}{5}\right)

d

\left(1.7, \,-2.8\right) and \left(-0.3,\,3\right)

7

Find the center of the circle.

a
-2
-1
1
2
3
4
5
6
x
-2
-1
1
2
3
4
5
6
y
b
-4
-3
-2
-1
1
2
3
x
-2
-1
1
2
3
4
5
y
8

Find the diameter of the circle using the center and a point on the circle.

a

Center: \left(8, -2\right)

Point on the circle: \left(2, 6\right)

b

Center: \left(-2, 3\right)

Point on the circle: \left(-6, 6\right)

9

Consider each of the following sets of endpoints of the diameters of a circle:

i
Find the center of the circle.
ii
Find the exact length of the diameter of the circle.
a

3\left(-1,3\right) and \left(4,-2\right)

b

\left(1, 1\right) and \left(-8, 4\right)

c

\left(2, -5\right) and \left(-3, 4\right)

d

\left(0, 0\right) and \left(6, 8\right)

e

\left(\dfrac{1}{2}, \dfrac{1}{3}\right) and \left(\dfrac{3}{2}, \dfrac{5}{3}\right)

f

\left(1.1, 2.2\right) and \left(3.4, 5.6\right)

10

A circle has a radius of \sqrt{13} units. Which of the following could represent the center and a point on the circle?

A

Center: \left(2, -2 \right)

Point on Circle: \left( 5, 2 \right)

B

Center: \left(0, -2 \right)

Point on Circle: \left( -3, 4 \right)

C

Center: \left(-6, -1 \right)

Point on Circle: \left( -3, -3 \right)

D

Center: \left(5, 6 \right)

Point on Circle: \left( -1, 2 \right)

11

Find the length of the radius of a circle when given the endpoints of its diameter. Round your answers to two decimal places.

a
Endpoints \left(0, 0\right) and \left(6, 8\right)
b
Endpoints \left(3, 4\right) and \left(7, 9\right)
c
Endpoints \left(5, 2\right) and \left(-3, -4\right)
d
Endpoints \left(-2, 3\right) and \left(4, -1\right)
e
Endpoints \left(\dfrac{1}{2}, \dfrac{1}{3}\right) and \left(\dfrac{3}{2}, \dfrac{5}{3}\right)
f
Endpoints \left(1.1, 2.2\right) and \left(3.4, 5.6\right)
g
Endpoints \left(0.4, 1.2\right) and \left(2.8, -0.8\right)
h
Endpoints \left(\dfrac{3}{2}, -\dfrac{5}{2}\right) and \left(-\dfrac{3}{2}, \dfrac{5}{2}\right)
12

Determine the coordinates of the center of the circle when given the endpoints of its diameter.

a
Endpoints \left(2, 4\right) and \left(-2, -4\right)
b
Endpoints \left(1, 2\right) and \left(3, 4\right)
c
Endpoints \left(-3, -5\right) and \left(1, 7\right)
d
Endpoints \left(\dfrac{1}{2}, \dfrac{3}{4}\right) and \left(\dfrac{5}{2}, \dfrac{7}{4}\right)
e
Endpoints \left(0, 5\right) and \left(6, 0\right)
f
Endpoints \left(-\dfrac{1}{3}, -\dfrac{2}{3}\right) and \left(\dfrac{2}{3}, \dfrac{1}{3}\right)
g
Endpoints \left(1.1, 2.2\right) and \left(3.4, 5.6\right)
h
Endpoints \left(-1.5, 2.5\right) and \left(4, -3.5\right)
13

Zoe claims the center of this circle is \left(4,3\right). Do you agree or disagree? How do you know?

-1
1
2
3
4
5
6
7
x
-1
1
2
3
4
5
6
7
y
14

A circle has a center at \left(-3, 6 \right) and a radius of 2\sqrt{5}. Select the graph of the circle.

A
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
x
-1
1
2
3
4
5
6
7
8
9
10
11
12
y
B
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
x
-1
1
2
3
4
5
6
7
8
9
10
11
12
y
C
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
x
-1
1
2
3
4
5
6
7
8
9
10
11
12
y
D
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
x
-1
1
2
3
4
5
6
7
8
9
10
11
12
y
15

A circle has a radius of 2\sqrt{30}. Select the graph of the circle.

A
-12
-10
-8
-6
-4
-2
2
4
6
8
10
12
x
-6
-4
-2
2
4
6
8
10
12
14
16
18
y
B
-8
-6
-4
-2
2
4
6
8
10
12
14
16
x
-8
-6
-4
-2
2
4
6
8
10
12
14
16
y
C
-12
-8
-4
4
8
12
x
-12
-8
-4
4
8
12
16
y
D
-10
-8
-6
-4
-2
2
4
6
8
10
x
-12
-10
-8
-6
-4
-2
2
4
6
8
y
Let's extend our thinking
16

The points \left(5, 3 \right) and \left(5, 7 \right) lie on the circumference of a circle. Which of the following could be the coordinates of the center of the circle?

A
\left(0, 4 \right)
B
\left(-1, 6 \right)
C
\left(4, 3 \right)
D
\left(4, 5 \right)
17

Given: Circle G

Center G: \left(-1, 3\right)

Radius: 8 units

Which point lies on circle G?

A

\left(0, 4\right)

B

\left(3, 7\right)

C

\left(-1, 11\right)

D

\left(-4, 3\right)

18

Determine whether the point \left(5, 4\right) is inside, outside, or on the circle with center \left(1, 2\right) and diameter of 6.

19

Find two potential points on the circle with center \left(4, 3\right) and radius 5.

20

Could the points \left(-4, 2\right) and \left(4, 2\right) both lie on a circle with center \left(1,1\right)? Explain why or why not.

21

Consider the circle with center at \left(3, 4\right).

A sector is created from point A to the center to another point on the circle. The area of this sector is approximately 18.4 \text{ cm}^{2}. Which of the following could be the other point on the sector?

A
\left(3, -2\right)
B
\left(8.3, 1.1\right)
C
\left(3, 10\right)
D
\left(-2, 7.2\right)
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
11
x
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
11
y
22

Telecommunication towers can be used to transmit cellular phone calls. A graph with units measured in kilometers shows towers at points \left(0,0\right), \, \left(0,4\right), and \left(-2,2\right). These towers have a range of about 3 kilometers.

a

Sketch a graph and locate the towers. Are there any locations that may receive calls from more than one tower? Explain your reasoning.

b

The center of City A is located at \left(-3, 2.5\right), and the center of City B is located at \left(4,7\right). Each city has a radius of 1.5 kilometers. Which city seems to have better cell phone coverage? Explain your reasoning.

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Outcomes

G.PC.4

The student will solve problems in the coordinate plane involving equations of circles.

G.PC.4bi

Solve problems in the coordinate plane involving equations of circles: i) given a graph or the equation of a circle in standard form, identify the coordinates of the center of the circle;

G.PC.4bii

Solve problems in the coordinate plane involving equations of circles: ii) given the coordinates of the endpoints of a diameter of a circle, determine the coordinates of the center of the circle.

G.PC.4biii

Solve problems in the coordinate plane involving equations of circles: iii) given a graph or the equation of a circle in standard form, identify the length of the radius or diameter of the circle.

G.PC.4biv

Solve problems in the coordinate plane involving equations of circles: iv) given the coordinates of the endpoints of the diameter of a circle, determine the length of the radius or diameter of the circle.

G.PC.4bv

Solve problems in the coordinate plane involving equations of circles: v) given the coordinates of the center and the coordinates of a point on the circle, determine the length of the radius or diameter of the circle; and

G.PC.4bvi

Solve problems in the coordinate plane involving equations of circles: vi) given the coordinates of the center and length of the radius of a circle, identify the coordinates of a point(s) on the circle.

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