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Springboard - Geometry
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2020F03EE2D4D2CC47D093CB67850C5BEABA
2020
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Andrew Lawrence's
SpringBoard Geometry Texas Edition 2015©Texas 2015
Grade 9
SpringBoard Algebra 1 Texas Edition 2014©TX 2014
Grade 10
SpringBoard Geometry Texas Edition 2015©TX 2015
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Lesson 27-1: Circles on the Coordinate Plane
Lesson 272
Completing the Square to Find the Center and Radius of a
Circle
Learning Targets
p. 396
Find the center and radius of a circle given its equation.
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Complete the square to write the equation of a circle in the form (x − h)2 + (y − k)2 = r2.
Vocabulary Organizer (Learning Strategy)
Definition
Using a graphic organizer to keep an ongoing record of vocabulary words with definitions, pictures,
notes, and connections between words
Purpose
Supports a systematic process of learning vocabulary
Create Representations (Learning Strategy)
Definition
Creating pictures, tables, graphs, lists, equations, models, and /or verbal expressions to interpret text
or data
Purpose
Helps organize information using multiple ways to present data and to answer a question or show a
problem solution
Think-Pair-Share (Learning Strategy)
Definition
Thinking through a problem alone, pairing with a partner to share ideas, and concluding by sharing
results with the class
Purpose
Help
Enables the development of initial ideas that are then tested with a partner in preparation for
revising ideas and sharing them with a larger group
Suggested Learning Strategies
Vocabulary Organizer, Create Representations, Think-Pair-Share
When an equation is given in the form (x − h)2 + (y − k)2 = r2, it is easy to identify the center and
radius of a circle, but equations are not always given in this form. For instance, the path of a
windmill is more realistically represented by a more complicated equation.
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Math Tip
To complete the square:
Keep all terms containing x on the left. Move the constant to the right.
If the x2 term has a coefficient, divide each term by that coefficient.
Divide the x-term coefficient by 2, and then square it. Add this value to both sides of the
equation.
Simplify.
Write the perfect square on the left.
1. Suppose the path of a windmill is given by the equation x2 + 6x + (y − 2)2 = 16. Describe how
this equation is different from the standard form of the circle equation.
In algebra, you learned how to complete the square to rewrite quadratic polynomials, such as x2 +
bx, as a factor squared. The process for completing the square is depicted below.
Help
2. What term do you need to add to each side of the equation x2 + 6x + (y − 2)2 = 16 to complete
the square for x2 + 6x?
3. Complete the square and write the equation in the form (x − h)2 + (y − k)2 = r2.
4. Make use of structure. Identify the center and the radius of the circle represented by your
equation in Item 3.
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Check Your Understanding
p. 397
5. Complete each square, and write the equation in the form (x − h)2 + (y − k)2 = r2.
a.
x
b.
x
2
2
− 2x + y
2
= 3
+ 3x + (y + 2)
2
= 1
Sometimes equations for circles become so complicated that you may need to complete the square
on both variables in order to write the equation in standard circle form.
Example A
A mural has been planned using a large coordinate grid. An artist has been asked to paint a red
circular outline around a feature according to the equation x2 − 4x + y2 + 10y = 6. Determine the
center and the radius of the circle that the artist has been asked to paint.
Follow these steps to write the equation in standard circle form.
Connect to AP
You can use what you know about circles to determine the equation for a sphere. In calculus,
you will learn to compute how fast the radius and other measurements of a sphere are changing
at a particular instant in time.
Step 1:
Write the equation.
Help
x2 − 4x + y2 + 10y = 7
Step 2:
Complete the square on both the x- and y-terms.
To complete the square on x2 − 4x, take half of 4 and square it. Add this term to both sides of the
equation. Then simplify, and write x2 − 4x + 4 as a perfect square.
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x
2
− 4x + 4 + y
(x − 2)
2
+ y
2
2
+ 10y
=
7 + 4
+ 10y
=
11
To complete the square on y2 +10, take half of 10 and square it. Add this term to both sides of the
equation. Then simplify, and write y2 + 10y + 25 as a perfect square.
(x − 2)
2
+ y
2
(x − 2)
+ 10y + 25
2
+ (y + 5)
2
=
11 + 25
=
36
Step 3:
Determine the center and radius of the circle.
Using the equation, the center of the circle is at (2, −5) and the radius is 6.
Try These A
Find the center and radius of each circle.
a. (x − 5)2 + y2 − 2y = 8
b. x2 + 4x + (y − 4)2 = 12
c. x2 − 8x + y2 − 14y = 16
d. x2 + 6x + y2 + 12y = 4
Check Your Understanding
Help
p. 398
6. Refer to Example A. Explain how you know the x-coordinate of the center of the circle is a
positive value and the y-coordinate is negative.
7. Circle Q is represented by the equation (x + 3)2 + y2 + 18y = 4. Bradley states that he needs to
add 18 to each side of the equation to complete the square on the y-term. Marisol disagrees
and states that it should be 9. Do you agree with either student? Justify your reasoning.
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8. Circle P is represented by the equation (x − 8)2 + 9 + y2 + 6y = 25.
a. What is the next step in writing the equation in standard form for a circle?
b. What is the radius of the circle?
Lesson 27-2 Practice
9. Determine if the equation given is representative of a circle. If so, determine the coordinates
of the center of the circle.
a. x2 + y2 − 6y = 16
b. x − 49 + y2 − 6y = 4
c. x2 − 2x + y2 + 10y = 0
10. Determine the center and radius of each circle.
a. (x − 6)2 + y2 − 8y = 0
b. x2 − 20x + y2 − 12y = 8
Help
c. x2 + 14x + y2 + 2y = 14
11. Reason quantitatively. On a two-dimensional diagram of the solar system, the orbit of Jupiter
is represented by circle J. Circle J is drawn on the diagram according to the equation x2 − 4x
− 9 + y2 + 5 = 0.
a. The center of the circle falls on which axis?
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b. What are the coordinates of the center of the circle?
c. What is the radius?
Activity 27 Practice
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