12.7 Circles in the Coordinate Plane
Objectives:
G.GPE.1: Derive the equation of a circle of given center and radius using the Pythagorean
Theorem; complete the square to find the center and radius of a circle given by an
equation.
For the board: You will be able to write the equation of a circle and to use the equation and its graph
to solve problems.
Anticipatory Set:
The equation of a circle is based on the Distance Formula
and the fact that all points on a circle are equidistant
from the center.
(x, y)
r
(h, k)
Equation of a Circle Theorem
The equation of a circle with center (h, k) and
radius r is (x – h)2 + (y – k)2 = r2.
Open the book to page 847 and read example 1.
Example: Write the equation of each circle.
A. circle J with center J(2, -2) and radius 4.
(x – 2)2 + (y + 2)2 = 16
B. circle K that passes through J(6, 4) and has center K(1, -8)
Find the radius by finding the distance between J and K.
Radius = 1  6   8  4 
(x – 1)2 + (y + 8)2 = 169
2
2
 52   12 2
 25  144  169  13
White Board Activity:
Practice: Write the equation of each circle.
A. circle Z with center Z(-4, 7) and radius 6.
(x + 4)2 + (y – 7)2 = 36
B. circle M with center M(-3, 5) passing through N(1, 2).
Find the radius by finding the distance between M and N.
Radius = 1  3  2  5 
(x + 3)2 + (y - 5)2 = 25
2
2
42   32
 16  9  25  5
Example: Determine the center and radius of the circle with the given equation.
(x – 3)2 + (y – 4)2 = 25
center (3, 4)
radius = 5
White Board Activity:
Practice: Determine the center and radius of the circle with the given equation.
(x + 6)2 + (y – 8)2 = 49
center (-6, 8)
radius = 7
To graph a circle:
1. Identify the center. Plot it.
3. Determine the radius. Locate 4 points on the circle by moving up, down, right, and left the length
of the radius from the center.
5. Sketch the circle.
Open the book to page 849 and study example 2.
Example: Graph each circle.
A. x2 + y2 = 16
center (0, 0)
radius = 4
2 units x 2 units
Graphing Activity:
Practice: Graph each circle.
A. x2 + y2 = 9
center (0, 0)
radius = 3
B. (x – 3)2 + (y + 4)2 = 9
center (3, -4)
radius = 3
2 units x 2 units
B. (x + 2)2 + (y – 1)2 = 25
center (-2, 1)
radius = 5
2 units x 2 units
Example: Write the equation of each circle. Each box is 2 units x 2 units.
Steps:
1. Determine the radius.
2. Count the distance from the center right/left/up/down. This will be the radius.
A.
B.
Center = (3, 3), radius = 5
(x – 3)2 + (y – 3)2 = 25
Center = (2, -2), radius = 6
(x – 2)2 + (y + 2)2 = 36
White Board Activity:
Practice: Write the equation of each circle. Each box is 2 units x 2 units.
A.
B.
Center = (0, 0), radius = 6
X2 + y2 = 36
Center = (-3, 3), radius = 5
(x + 3)2 + (y – 3)2 = 25
All circles are similar because they have the same shape.
Similarity transformations can be used to prove this.
Steps:
(1). Plot the center of the first circle and use the radius to plot 4 points to assist in drawing the
circle. (up, down, right, left)
(2). Plot the center of the second circle and use the radius to plot 4 points to assist in drawing
the circle. (up, down, right ,left)
(3). Find the scale factor. (radius 2nd circle/radius 1st circle)
(4). Use the scale factor to find the image of the 1st circle)
(5). Determine the translation which will move the image circle onto the 2nd circle.
(x, y) → (x + h, y + k).
Open the book to page 474 and read example 3.
Example: Prove each statement.
a. Circle A with center (0, 0) and radius 1
is similar to circle B with center (6, 0)
and radius 3.
Dilation with a scale factor of 3
followed by a translation 6 units right.
b. Circle C with center (0, -3) and radius 2
is similar to circle D with center (5, 1)
and radius 5.
Dilation with a scale factor of 2.5
followed by a translation right 5 and up 4.
Graphing Activity:
Practice: Prove that circle A with center (2, 1) and
radius 4 is similar to circle B with center
(-1, -1) and radius 2.
Dilation with a scale factor of ½ .
A pply the dilation factor to circle A
Getting the green circle.
Translate the green circle down 2 and
Left 3 getting circle B.
Assessment:
Question student pairs.
Independent Practice:
Text: pgs. 850 – 853 prob. 1 – 8, 10 – 17, 19 – 21, 30 – 32;
pgs. 477 – 478 prob. 11, 12, 22.
For a Grade:
Text: pgs. 850 – 853 prob. 12, 14, 20; pg. 478 prob. 22.
A
B