32 Equation of a Circle Directed Round and Round Lesson 32 ...(x 2+ 3)2 + y = 10 10. Determine the series of transformations that map the circle (x − 2)2 + (y + 5) 2 = 1 to the circle

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Equation of a CircleRound and RoundLesson 32-1 Circles on the Coordinate Plane

Learning Targets: • Derive the general equation of a circle given the center and radius.• Write the equation of a circle given three points on the circle.

SUGGESTED LEARNING STRATEGIES: Quickwrite, Think-Pair-Share, Create Representations, Vocabulary Organizer

A windmill converts wind energy to power when its blades rotate around its center. Each blade represents the radius of the windmill’s circular path. How can this path be described mathematically? 1. Suppose that the coordinate plane is positioned so that the center of a

windmill is at point (2, 3). The blade of the windmill is 5 decameters long.

Let (x, y) be any point that lies on the tip of this blade.

x

5

(x, y)

(2, 3)

The equation of the circular path of the tip of the blade represents all points (x, y) that lie a fixed distance of 5 decameters from a given point, (2, 3). You can use the Pythagorean theorem (or the Distance Formula) to find the equation of this circle. a. Draw a right triangle so that the hypotenuse is the radius from the

center of the circle to (x, y). Use a vertical segment and a horizontal segment.

b. What is the length of the horizontal leg of the right triangle in terms of x?

c. What is the length of the vertical leg of the right triangle in terms of y?

d. Use the Pythagorean theorem to write the equation of the circle.

A wind farm is an array of wind turbines, or modern-day windmills used for generating electrical power. One of the world’s largest wind farms at Altamont Pass, California, consists of nearly 5,000 wind turbines. Wind farms supply about 5% of California’s electricity needs.

STEMCONNECT TO

As with linear and quadratic equations, the solutions (x, y) that satisfy the equation represent the points of the graph on the coordinate plane.

MATH TIP

A decameter is a metric unit of length equal to 10 meters. The prefix deca- means 10.

ACADEMIC VOCABULARY

See above for possible triangle.

x − 2

y − 3

(x − 2)2 + (y − 3)2 = 52

Activity 32 • Equation of a Circle 479

ACTIVITY 32

Academic Vocabulary

Common Core State Standards for Activity 32

HSG-GPE.A.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.

HSG-GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2).

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ACTIVITY 32

Directed

Activity Standards FocusIn this activity, students derive standard equations for circles using the Pythagorean theorem. They use the technique of completing the square to write an equation of a circle in the form(x − h)2 + (y − k)2 = r2 in order to identify the center and radius of the circle.

Lesson 32-1

PLAN

Pacing: 1 class periodChunking the Lesson#1 #2 #3–4 #5–7Check Your Understanding#11 #12Check Your UnderstandingLesson Practice

TEACH

Bell-Ringer ActivityOn the board, plot A(−2, 5) and B(3, 8) on a coordinate plane and connect the two points. Draw right triangle ABC with segment AB as the hypotenuse, and ask students to find its length. It may be helpful for students to determine the coordinates of point C. This will help them understand that the length of the horizontal leg is the difference between the x-coordinates of points A and C, and the length of the vertical leg is the difference between the y-coordinates of points B and C. This understanding will be useful throughout the lesson, particularly with Items 1b and 1c.

1 Create Representations The purpose of this item is to use the Pythagorean theorem to write the equation of a circle with a center not at the origin. For Item 1e, it may be helpful to remind students that a point lies on the graph of an equation if substituting the x- and y- values of the point produces a true statement.

TEACHER to TEACHER

CLASSROOM-TESTED TIPConnect several arbitrary points (x, y) to the center of the circle to create several right triangles. Demonstrate that using the Pythagorean theorem results in the same equation for the circle.

Students may be familiar with the term decathlon, which consists of ten track and field events. This will serve as a mnemonic help for students to remember the term decameters.


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Lesson 32-1Circles on the Coordinate Plane

Since each point on the circle must be 5 decameters from the center, your equation must be true for all points on the circle.

e. Use your equation from part d to determine if the following points lie on the circular path of the tip of the blade.i. (2, 7) ii. (7, 3) iii. (–2, 0)

Consider a circle that has its center at the point (h, k) and has radius r.

x

r

y

(x, y)

(h, k)

2. Suppose (x, y) is any point on the circle. Use the Pythagorean theorem to write the equation of this circle as (x − h)2 + (y − k)2 = r2. a. Draw a right triangle so that the hypotenuse is the radius from the


b. Claire says that the lengths of the horizontal and vertical legs of the triangle are equal to x − h and y − k, respectively. Explain why one of the expressions is incorrect for the diagram shown.

c. Use absolute value to rewrite both of Claire’s expressions for the lengths of the legs. Explain why the expressions work regardless of the position of (x, y).

d. Use the Pythagorean theorem and your expressions from part c to write the equation of the circle.

e. The center-radius form of a circle is (x − h)2 + (y − k)2 = r2, where (h, k) is the center of the circle and r is the radius. Justify why this form is equivalent to your equation from part d.

Geometry and Algebra In Lesson 30-1, you learned that the definition of a circle is the set of all points in a plane that lie a given distance from a fixed point in the plane. You now are deriving the algebraic representation of the circle.

POINT OF INTEGRATION

Yes, it lies on the circle.Yes, it lies on the circle.No, it does not lie on the circle.

See above for possible triangle.

The expression x − h is incorrect, because its value is negative. A distance cannot be negative.

x h y k − −2 2 2+ = r

x h − and y k − ; Taking the absolute values of the expressions will guarantee that the lengths are positive.

Possible answer: The expressions x h − and y k − are squared in the formula, and the resulting values are nonnegative, so the absolute value signs are not needed.

480 SpringBoard® Integrated Mathematics II, Unit 6 • Extending Two Dimensions to Three Dimensions

continuedcontinuedcontinuedACTIVITY 32

479-488_SB_IM2_SE_U06_A32.indd Page 480 3/15/16 4:02 PM user /112/SB00012_DEL/work/indd/SE/IM2/Application_files/SE_IM2_Unit_06/SB_IM2_SE_U06_ ...

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ACTIVITY 32 Continued

2 Shared Reading, Marking the Text Create Representations In Item 2, students confirm the standard equation for a circle. They also consider the meaning of (x − h) and (y − k), in that either of these expressions might result in a negative quantity, but the square of the expressions always result in a positive quantity. Monitor students’ discussions to make sure that they understand each item. As students work on Item 2b, ask them whether the expressions (x − h) and (y − k) result in positive or negative values as (x, y) is in each of the four quadrants.

Geometry and Algebra It is important for students to connect the definition of a circle with the equation of a circle. Students must understand that (x, y) is not a fixed point, but the set of all points that lie a given distance (the length of the radius) from the fixed point (the center of the circle). Students will see that the equation will be true for any point (x, y) that lies on the circle.



MINI-LESSON: Using the Distance Formula

If students need a review of using the distance formula to find distances in the coordinate plane, a mini-lesson is available to provide practice.

See the Teacher Resources on SpringBoard Digital for a student page for this mini-lesson.

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Check Your Understanding

8. Write the equation of the circle described. a. center (0, 0) and radius 7 b. center (0, 2), radius = 10 c. center (9, −4), radius = 15

9. Identify the center and radius of the circle. a. (x − 2)2 + (y + 1)2 = 9 b. (x + 3)2 + y2 = 10

10. Determine the series of transformations that map the circle (x − 2)2 + (y + 5)2 = 1 to the circle x2 + (y − 7)2 = 64.

3. Reason quantitatively. Write the equation of a circle in center-radius form with center (−4, −3) and radius r. How is the center-radius form of a circle affected when the x- and y-coordinates of the center are both negative?

4. Determine if the equation represents a circle. If it does, write the center and radius. a. (x − 3)2 + y2 = 100

b. (x − 1)2 − (y − 7)2 = 25

c. 2(x − 8)2 + 2(y + 4)2 = 32

5. Construct viable arguments. Consider any two circles in the coordinate plane. Which two transformations can be used to map one circle onto the other? When is only one transformation needed? Explain.

6. Describe a series of transformations to map the circle x2 + y2 = 4 to the circle from Item 4a.

7. Express regularity in repeated reasoning. Write a series of transformations to map the circle from Item 4a to the circle x2 + y2 = 4.

Remember:

(a − b)2 = (b − a)2

MATH TIP

Possible answer: (x − (−4))2 + (y − (−3))2 = r2, or (x + 4)2 + (y + 3)2 = r2; The values of h and k will be negative, so the equation may be written with sums rather than differences.

Yes; center (3, 0) and radius 10

No

Yes; center (8, −4) and radius 4

Translate 3 units right, then dilate about (3, 0) by scale factor 252

12 5= . .

Translate 3 units left, then dilate about (0, 0) by scale factor 225

0 08= . .

A translation and a dilation; translate the center of one circle to the center of the other, and then dilate about the center. If the circles already share a center, or have the same radius, then only one transformation is needed.



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3–4 Create Representations, Think-Pair-Share, Quickwrite Students confirm their understanding of the standard form of the equation of a circle by writing an equation with a center in the third quadrant. Monitor students’ work to make sure that they correctly change signs in (x − h) and (y − k) for the center given. Students further confirm their understanding by determining if given equations represent circles. Particularly observe students’ responses to Item 4b, making sure that they focus on the negative sign between (x − 1)2 and (y − 7)2. It may help to use the graphing calculator on SpringBoard Digital to show them how the graph changes when there is a negative sign instead of a positive sign between those two expressions.

5–7 Think-Pair-Share, Quickwrite Students connect the equation of a circle with translations. If students are having trouble with Item 5, have them draw two circles on the coordinate plane. The circles should have different centers and different radii. Have students explain how one of these circles can be transformed to map onto the second circle. Then students can generalize this explanation.

Check Your UnderstandingDebrief students’ answers to these items to confirm that they can write the equation given center and radius, and find the center and radius given the equation. Pay special attention to Item 8c, in which students start with a negative y-coordinate. This item confirms that students understand what to do with a square root form used as the radius.

Answers 8. a. x2 + y2 = 49 b. x2 + (y −2)2 = 100 c. (x − 9)2 + (y + 4)2 = 15 9. a. center (2, −1); radius = 3 b. center (−3, 0); radius = 10 10. Translate 2 units left and 12 units

up, and then dilate by a factor of 8.


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Suppose the center of the windmill is positioned at (−2, −3). 11. Draw a circle with center (−2, −3) that contains the point (1, 1).

a. Write the equation of any circle with radius r and center (−2, −3).

b. Substitute (1, 1) for (x, y) in the equation from part a. Explain what information this gives about the circle.

c. Write the equation of the circle with center (−2, −3) that contains the point (1, 1).

12. Consider the circle that has a diameter with endpoints (−1, 4) and (9, −2). a. Determine the midpoint of the diameter. What information does this

give about the circle?

b. Using one of the endpoints of the diameter and the center of the circle, write the equation of the circle.

c. Using the other endpoint of the diameter and the center of the circle, write the equation of the circle.

d. What do you notice about your responses to parts b and c above? What does that tell you about writing the equation of a circle given the endpoints of the diameter?

− + + −

= ( )1 9

24 2

24 1, ( ) , is the center of the circle.

Using the endpoint (−1, 4): ( −1 − 4)2 + (4 − 1)2 = r2; (−5)2 + (3)2 = r2; and (x − 4)2 + (y −1)2 = 34.

Using the endpoint (9, −2): (9 − 4)2 + (−2 − 1)2 = r2; (5)2 + (−3)2 = r2; (x − 4)2 + (y − 1)2 = 34

Sample answer: The responses to parts b and c are the same. It does not matter which endpoint you use to write the equation.

(x + 2)2 + (y + 3)2 = r2

(1 + 2)2 + (1 + 3)2 = r2; (3)2 + (4)2 = r2; 25 = r2; and 5 = radius of the circle

(x + 2)2 + (y + 3)2 = 25



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11 Think-Pair-Share, Create Representations, Quickwrite The purpose of this item is to have students build the equation of a circle and use the equation to find r2 and then write the value of the radius. Students should be asked to write the answer to each part independent of their collaborative group before comparing answers with their group.

12 Think-Pair-Share, Create Representations, Quickwrite Students extend their ability to write the equations of a circle to more complicated situations. They are given the endpoints of the diameter and must find the center of the circle.


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13. Explain why the equations (x − 1)2 + (y − 3)2 = 16 and (1 − x)2 + (3 − y)2 = 16 represent the same circle.

14. Write the equation of the circle described. a. center (3, 6) passes through the point (0, −2) b. diameter with endpoints (2, 5) and (−10, 7)

LESSON 32-1 PRACTICE 15. Write the equation of a circle centered at the origin with the given

radius. a. radius = 6 b. radius = 12

16. Write the equation of a circle given the center and radius. a. center: (7, 2); radius = 5 b. center: (−4, −2); radius = 9

17. Identify the center and radius of the circle. a. (x − 5)2 + y2 = 225 b. (x + 4)2 + (y − 2)2 = 13

18. Write a series of transformations to map the circle from Item 16a to Item 17a.

19. Two circles on the coordinate plane have radii R and r, with R > r. They share the same center. What is the transformation that maps the smaller circle to the larger circle?

20. Model with mathematics. An engineer draws a cross-sectional area of a pipe on a coordinate plane with the endpoints of the diameter at (2, −1) and (−8, 7). Determine the equation of the circle determined by these endpoints.



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Check Your UnderstandingDebrief students’ answers to these items to confirm their understanding of writing the equation of a circle given a variety of situations.

Answers 13. (x − 1) and (1 − x) are opposites,

and (y − 3) and (3 − y) are opposites. When squared, the opposites are equivalent.

14. a. (x − 3)2 + (y − 6)2 = 73 b. (x + 4)2 + (y − 6)2 = 37

ASSESS

Students’ answers to the Lesson Practice items will provide a formative assessment of their understanding of writing the equation of a circle, and of students’ ability to apply their learning.Short-cycle formative assessment items for Lesson 32-1 are also available in the Assessment section on SpringBoard Digital.Refer back to the graphic organizer the class created when unpacking Embedded Assessment 2. Ask students to use the graphic organizer to identify the concepts or skills they learned in this lesson.

ADAPT

Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing equations of circles. Students who may need more practice with the concepts in this lesson may benefit from deriving the equation of a circle centered at the origin with a radius of 2. Students should understand that there is no need to subtract values from x and y because these values already represent the horizontal and vertical distances from the center. Once students understand how the equation of this circle was derived, they can go on to derive the equations of circles that are not centered at the origin. Students who may need more practice identifying the center and radius of a circle given its equation in center-radius form should write (x − h)2 + (y − k)2 = r 2 above the given equation. The equations should line up so that it is easy for the student to identify the center and radius within the equation on which they are working.See the Activity Practice on page 487 and the Additional Unit Practice in the Teacher Resources on SpringBoard Digital for additional problems for this lesson. You may wish to use the Teacher Assessment Builder on SpringBoard Digital to create custom assessments or additional practice.

LESSON 32-1 PRACTICE 15. a. x2 + y2 = 36 b. x2 + y2 = 12 16. a. (x − 7)2 + (y − 2)2 = 25 b. (x + 4)2 + (y + 2)2 = 81 17. a. center: (5, 0); radius: 15 b. center: (−4, 2); radius: 13 18. Translate 2 units down and 2 units

left, and then dilate by a factor of 3. 19. dilate the smaller circle by a factor

of Rr

. 20. (x + 3)2 + (y − 3)2 = 41


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Lesson 32-2Completing the Square to Find the Center and Radius of a Circle

Learning Targets:• Find the center and radius of a circle given its equation.• Complete the square to write the equation of a circle in the form

(x − h)2 + (y − k)2 = r2.

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. 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.

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.

x

x x2x2 bxbx++ x

xbb2

b2

x + b2

x + b2

b2( )

2

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.

MATH TIP

Geometry and AlgebraThe equation of a circle in center-radius form can be compared to the equation of a parabola in vertex form. Writing equations in these forms can reveal properties of the circle or the parabola’s geometric shape.


It includes the terms x2 and 6x.

(x + 3)2 + (y − 2)2 = 25

The center is at (−3, 2). The radius is 5.

9



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Lesson 32-2

PLAN

Pacing: 1 class periodChunking the Lesson#1–4Check Your UnderstandingExample ACheck Your UnderstandingLesson Practice

TEACH

Bell-Ringer ActivityWrite x2 − 4x + 4 + y2 = 36 and x2 + y2 − 10y + 25 = 100 on the board. Ask students to identify the center and radius of both equations. Most students will recognize that they will need to write x2 − 4x + 4 as a squared binomial and write y2 − 10y + 25 as a squared binomial to find the radius of each circle. This understanding is an important step in understanding the underlying process of completing the square.

1–4 Vocabulary Organizer, Create Representations, Think-Pair-Share In Items 1–4, students review the process they learned in algebra for completing the square and apply it to the equation of a circle. The visual following Item 1 helps students recall that the constant to be added is the square of half of the x-term. For Item 3, you may have to remind students to add the constant to each side of the equation, recalling the Addition Property of Equality.

Geometry and Algebra Encourage students to recall their study of parabolas and compare their understanding of how finding the vertex of a parabola is similar to finding the center of a circle. In their future study of mathematics, students will apply completing the square to find the centers of ellipses and hyperbolas.


MINI-LESSON:

Students may benefit from practicing how to rewrite a quadratic function by completing the square. This skill will be useful when students rewrite the equations of circles by completing the square.

See the Teacher Resources on SpringBoard Digital for a student page for this mini-lesson.

Rewriting a Quadratic Function by Completing the Square


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5. Complete each square, and write the equation in the form (x − h)2 + (y − k)2 = r2. a. x x y2 22 3− + = b. x x y2 23 2 1+ + + =( )

Example AA 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.Step 1: Write the equation.

x2 − 4x + y2 + 10y = 7Step 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.

x2 − 4x + 4 + y2 + 10y = 7 + 4 (x − 2)2 + y2 + 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 + y2 + 10y + 25 = 11 + 25 (x − 2)2 + (y + 5)2 = 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 AFind 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

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.

APCONNECT TO

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.

center: (5, 1); radius: 3

center: (−2, 4); radius: 4

center: (4, 7); radius: 9

center: (−3, −6); radius: 7



APTOCONNECT

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Check Your UnderstandingDebrief this section by confirming that students have properly completed the square and rewritten the equations in (x − h)2 + (y − k)2 = r2 form. To confirm their understanding, ask them to state the center and radius of each circle. Pay special attention to part b. Students must recognize that since the signs between the two terms in parentheses are positive, the result is negative coefficients for the x- and y-coordinates of the center. You may wish to demonstrate this by showing

that x +( )32

2can be written as

x − −( )

32

2, so that it is easier to see

that h = − 32

. Similarly, (y + 2)2 can be

written as (y − (−2))2, so k = −2.

Answers5. a. ( ) ( )x y− + − =1 0 42 2

b. x y+( ) + + =32

2 134

22( )

Example A Shared Reading, Marking the Text This example allows students to see a situation in which both the x-terms and the y-terms require completing the square to rewrite the equation in (x − h)2 + (y − k)2 = r2 form. After reading the initial problem, ask students how the equation is different from the ones they have worked with thus far. Look for the analysis that both x and y have squared and first-degree terms. So, both terms require completing the square. As you work through Step 2, have students mark the completed constant for x and then the completed constant for y, and mark that these are added to both sides of the equation. Some students will want to complete the square for the x-term and the y-term in the same step. Clearly, they can do this, but they must be very careful to think of the two parts distinctly and to add to the right side each “completed” value.

Try These A Think-Pair-Share, Critique Reasoning These items will confirm students’ understanding of Example A. Have students compare their answers, and if they don’t agree, compare their work to identify any error in reasoning.

Students’ understanding of circles will be central to their future work, especially in understanding trigonometric concepts.


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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.

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 32-2 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 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? b. What are the coordinates of the center of the circle? c. What is the radius?



TEACHER to TEACHER

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Check Your UnderstandingDebrief this section by confirming that students can explain their reasoning. Item 6 requires students to explain how they can tell the signs of the coordinates of the center of the circle from an equation, Item 7 requires them to critique reasoning, and Item 8 requires them to think about the steps used to solve a problem.

CLASSROOM-TESTED TIPFrom your observations of students’ work in the first part of this lesson, identify those students with a firm grasp of completing the square. Group students so that at least one such student is in each group. As you monitor groups, ask, “How do you know?” so that group members must explain their understanding and answers to the Check Your Understanding items.

Answers6. The standard form of the equation of

a circle is (x − h)2 + (y − k)2 = r2. In the equation (x − 2)2 + (y + 5)2 = 36, h is positive and k is negative.

7. No; to complete the square, you take12 of 18 and square it. So, you should add 81 to both sides of the equation.

8. a. Use the Commutative Property to rewrite the equation as(x − 8)2 + y2 + 6y + 9 = 25, and then you can rewrite it as(x − 8)2 + (y + 3)2 = 25.

b. 5

ASSESS

Students’ answers to the Lesson Practice items will provide a formative assessment of their understanding of completing the square to find the center and radius of a circle, and of students’ ability to apply their learning.Short-cycle formative assessment items for Lesson 32-2 are also available in the Assessment section on SpringBoard Digital.Refer back to the graphic organizer the class created when unpacking Embedded Assessment 2. Ask students to use the graphic organizer to identify the concepts or skills they learned in this lesson.

LESSON 32-2 PRACTICE 9. a. yes; (0, 3) b. not representative of a circle c. yes; (1, −5) 10. a. center: (6, 4); radius: 4 b. center: (10, 6); radius: 12 c. center: (−7, −1); radius: 8 11. a. x-axis b. (−2, 0) c. radius = 8

ADAPT

Check students’ answers to the Lesson Practice to ensure that they understand how to complete the square to find the center and radius of a circle. Students who may need more practice with the concepts in this lesson should use the pattern of steps shown in Example A. In Item 11, it may be easier for students to start by moving the constants to the right side of the equation and then complete the square.See the Activity Practice on page 487 and the Additional Unit Practice in the Teacher Resources on SpringBoard Digital for additional problems for this lesson.You may wish to use the Teacher Assessment Builder on SpringBoard Digital to create custom assessments or additional practice.


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Equation of a CircleRound and Round

ACTIVITY 32 PRACTICEAnswer each item. Show your work.

Lesson 32-1 1. A circle is drawn on the coordinate plane to

represent an in-ground swimming pool. The circle is represented by the equation (x − 3)2 + y2 = 4. Which of the following points lies on the circle? A. (1, 0) B. (2, 1) C. (1, −2) D. (3, 0)

2. Two points on the circle given by the equation (x − 1)2 + (y +3)2 = 100 have the x-coordinate −7.What are the y-coordinates of those two points?

3. Write an equation for the circle described. a. center (−3, 0) and radius = 7 b. center (4, 3), tangent to the y-axis c. center (2, −1) and contains the point (4, 5) d. diameter, with endpoints (2, −5) and (4, 1)

4. Explain how you can tell by looking at the equation of the circle that the center of the circle lies on the y-axis.

5. Write the equation of a circle centered at the origin with the given radius. a. radius = 3 b. radius = 11

6. Which equation represents a circle with a diameter at endpoints (−1, 5) and (3, 7)? A. (x − 1)2 + (y − 6)2 = 5 B. (x − 2)2 + (y − 10)2 = 4 C. (x + 6)2 + (y + 1)2 = 25 D. (x + 10)2 + (y − 2)2 = 10

7. Which circle has a center of (0, –4)? A. (x − 4)2 + y2 = 4 B. (x + 1)2 + (y − 4)2 = 4 C. x2 + (y + 4)2 = 16 D. (x + 4)2 + (y − 4)2 = 16

8. Find a series of transformations that maps the circle (x − 2)2 + (y + 1)2 = 1 to the circle (x + 3)2 + y2 = 1.

9. A circle with radius 3 is centered at the origin. It is translated 2 units right and 3 units down, then dilated by a factor of 2. What is the equation for the transformed circle?

10. Identify the center and radius of the circle given by each equation. a. (x − 3)2 + y2 = 1 b. (x + 2)2 + (y + 1)2 = 16



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ACTIVITY PRACTICE 1. A 2. y = 3 or y = −9 3. a. (x + 3)2 + y2 = 49 b. (x − 4)2 + (y − 3)2 = 16 c. (x − 2)2 + ( y + 1)2 = 40 d. (x − 3)2 + ( y + 2)2 = 10 4. An x-term does not exist, only x2. 5. a. x2 + y2 = 9 b. x2 + y2 = 11 6. A 7. C 8. Translate 5 units left and 1 unit up. 9. (x − 2)2 + (y + 3)2 = 36 10. a. center (3, 0); radius = 1 b. center (−2, −1); radius = 4


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Lesson 32-2 11. Identify the center and radius of the circle

represented by the equation(x + 4)2 + y2 − 10y = 11.

12. Circle Q is represented by the equation (x + 1)2 + y2 + 6y = 5. Howard states that the center of the circle lies in Quadrant III. Do you agree? Justify your reasoning.

13. Circle P is represented by the equation (x − 11)2 + y2 + 10y = 25. What is the radius of the circle? A. 5 B. 10 C. 25 D. 50

14. Determine if the equation given is representative of a circle. If it is, determine the coordinates of the center of the circle. a. x + 5 − y2 − y = 5 b. x2 − 2 + y2 − 8y = 7 c. x2 + 2x + y2 − 16y = 0

15. For any equations in Item 14 that are not equations of a circle, explain how you determined your answer.

MATHEMATICAL PRACTICESLook For and Make Use of Structure

16. Given an equation of a circle, how can you determine when you need to complete the square to determine the coordinates of the center of the circle? Write an example of an equation of a circle where you would have to complete the square to determine that the circle has a radius of 10 and a center of (4, 6).




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11. center: (−4, 5); radius: 6 12. Yes; the center of the circle is at

(−1, −3), which is in Quadrant III. 13. D 14. a. no b. (0, 4) c. (−1, 8) 15. Sample answer: In Item 15a, there is

no x2-term, so the equation is not that of a circle.

16. Sample answer: When the equation does not show a squared binomial for the x- and y-terms; x2 − 8x + y2 − 12y = 48

ADDITIONAL PRACTICEIf students need more practice on the concepts in this activity, see the Additional Unit Practice in Teacher Resources on SpringBoard Digital for additional practice problems.


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Equation of a CircleRound and RoundLesson 32-1 Circles on the Coordinate Plane

Learning Targets: • Derive the general equation of a circle given the center and radius.• Write the equation of a circle given three points on the circle.

SUGGESTED LEARNING STRATEGIES: Quickwrite, Think-Pair-Share, Create Representations, Vocabulary Organizer

A windmill converts wind energy to power when its blades rotate around its center. Each blade represents the radius of the windmill’s circular path. How can this path be described mathematically? 1. Suppose that the coordinate plane is positioned so that the center of a

windmill is at point (2, 3). The blade of the windmill is 5 decameters long.

Let (x, y) be any point that lies on the tip of this blade.

x

5

(x, y)

(2, 3)

The equation of the circular path of the tip of the blade represents all points (x, y) that lie a fixed distance of 5 decameters from a given point, (2, 3). You can use the Pythagorean theorem (or the Distance Formula) to find the equation of this circle. a. Draw a right triangle so that the hypotenuse is the radius from the


b. What is the length of the horizontal leg of the right triangle in terms of x?

c. What is the length of the vertical leg of the right triangle in terms of y?

d. Use the Pythagorean theorem to write the equation of the circle.

A wind farm is an array of wind turbines, or modern-day windmills used for generating electrical power. One of the world’s largest wind farms at Altamont Pass, California, consists of nearly 5,000 wind turbines. Wind farms supply about 5% of California’s electricity needs.

STEMCONNECT TO

As with linear and quadratic equations, the solutions (x, y) that satisfy the equation represent the points of the graph on the coordinate plane.

MATH TIP

A decameter is a metric unit of length equal to 10 meters. The prefix deca- means 10.

ACADEMIC VOCABULARY


ACTIVITY 32

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Since each point on the circle must be 5 decameters from the center, your equation must be true for all points on the circle.

e. Use your equation from part d to determine if the following points lie on the circular path of the tip of the blade.i. (2, 7) ii. (7, 3) iii. (–2, 0)

Consider a circle that has its center at the point (h, k) and has radius r.

x

r

y

(x, y)

(h, k)

2. Suppose (x, y) is any point on the circle. Use the Pythagorean theorem to write the equation of this circle as (x − h)2 + (y − k)2 = r2. a. Draw a right triangle so that the hypotenuse is the radius from the


b. Claire says that the lengths of the horizontal and vertical legs of the triangle are equal to x − h and y − k, respectively. Explain why one of the expressions is incorrect for the diagram shown.

c. Use absolute value to rewrite both of Claire’s expressions for the lengths of the legs. Explain why the expressions work regardless of the position of (x, y).

d. Use the Pythagorean theorem and your expressions from part c to write the equation of the circle.

e. The center-radius form of a circle is (x − h)2 + (y − k)2 = r2, where (h, k) is the center of the circle and r is the radius. Justify why this form is equivalent to your equation from part d.

Geometry and Algebra In Lesson 30-1, you learned that the definition of a circle is the set of all points in a plane that lie a given distance from a fixed point in the plane. You now are deriving the algebraic representation of the circle.




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8. Write the equation of the circle described. a. center (0, 0) and radius 7 b. center (0, 2), radius = 10 c. center (9, −4), radius = 15

9. Identify the center and radius of the circle. a. (x − 2)2 + (y + 1)2 = 9 b. (x + 3)2 + y2 = 10

10. Determine the series of transformations that map the circle (x − 2)2 + (y + 5)2 = 1 to the circle x2 + (y − 7)2 = 64.

3. Reason quantitatively. Write the equation of a circle in center-radius form with center (−4, −3) and radius r. How is the center-radius form of a circle affected when the x- and y-coordinates of the center are both negative?

4. Determine if the equation represents a circle. If it does, write the center and radius. a. (x − 3)2 + y2 = 100

b. (x − 1)2 − (y − 7)2 = 25

c. 2(x − 8)2 + 2(y + 4)2 = 32

5. Construct viable arguments. Consider any two circles in the coordinate plane. Which two transformations can be used to map one circle onto the other? When is only one transformation needed? Explain.

6. Describe a series of transformations to map the circle x2 + y2 = 4 to the circle from Item 4a.

7. Express regularity in repeated reasoning. Write a series of transformations to map the circle from Item 4a to the circle x2 + y2 = 4.

Remember:

(a − b)2 = (b − a)2

MATH TIP



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Suppose the center of the windmill is positioned at (−2, −3). 11. Draw a circle with center (−2, −3) that contains the point (1, 1).

a. Write the equation of any circle with radius r and center (−2, −3).

b. Substitute (1, 1) for (x, y) in the equation from part a. Explain what information this gives about the circle.

c. Write the equation of the circle with center (−2, −3) that contains the point (1, 1).

12. Consider the circle that has a diameter with endpoints (−1, 4) and (9, −2). a. Determine the midpoint of the diameter. What information does this

give about the circle?

b. Using one of the endpoints of the diameter and the center of the circle, write the equation of the circle.

c. Using the other endpoint of the diameter and the center of the circle, write the equation of the circle.

d. What do you notice about your responses to parts b and c above? What does that tell you about writing the equation of a circle given the endpoints of the diameter?



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13. Explain why the equations (x − 1)2 + (y − 3)2 = 16 and (1 − x)2 + (3 − y)2 = 16 represent the same circle.

14. Write the equation of the circle described. a. center (3, 6) passes through the point (0, −2) b. diameter with endpoints (2, 5) and (−10, 7)

LESSON 32-1 PRACTICE 15. Write the equation of a circle centered at the origin with the given

radius. a. radius = 6 b. radius = 12

16. Write the equation of a circle given the center and radius. a. center: (7, 2); radius = 5 b. center: (−4, −2); radius = 9

17. Identify the center and radius of the circle. a. (x − 5)2 + y2 = 225 b. (x + 4)2 + (y − 2)2 = 13

18. Write a series of transformations to map the circle from Item 16a to Item 17a.

19. Two circles on the coordinate plane have radii R and r, with R > r. They share the same center. What is the transformation that maps the smaller circle to the larger circle?

20. Model with mathematics. An engineer draws a cross-sectional area of a pipe on a coordinate plane with the endpoints of the diameter at (2, −1) and (−8, 7). Determine the equation of the circle determined by these endpoints.



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Learning Targets:• Find the center and radius of a circle given its equation.• Complete the square to write the equation of a circle in the form

(x − h)2 + (y − k)2 = r2.

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. 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.

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.

x

x x2x2 bxbx++ x

xbb2

b2

x + b2

x + b2

b2( )

2

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.

MATH TIP

Geometry and AlgebraThe equation of a circle in center-radius form can be compared to the equation of a parabola in vertex form. Writing equations in these forms can reveal properties of the circle or the parabola’s geometric shape.




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5. Complete each square, and write the equation in the form (x − h)2 + (y − k)2 = r2. a. x x y2 22 3− + = b. x x y2 23 2 1+ + + =( )

Example AA 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.Step 1: Write the equation.

x2 − 4x + y2 + 10y = 7Step 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.

x2 − 4x + 4 + y2 + 10y = 7 + 4 (x − 2)2 + y2 + 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 + y2 + 10y + 25 = 11 + 25 (x − 2)2 + (y + 5)2 = 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 AFind 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

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.

APCONNECT TO

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.



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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.

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 32-2 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 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? b. What are the coordinates of the center of the circle? c. What is the radius?



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ACTIVITY 32 PRACTICEAnswer each item. Show your work.

Lesson 32-1 1. A circle is drawn on the coordinate plane to

represent an in-ground swimming pool. The circle is represented by the equation (x − 3)2 + y2 = 4. Which of the following points lies on the circle? A. (1, 0) B. (2, 1) C. (1, −2) D. (3, 0)

2. Two points on the circle given by the equation (x − 1)2 + (y +3)2 = 100 have the x-coordinate −7.What are the y-coordinates of those two points?

3. Write an equation for the circle described. a. center (−3, 0) and radius = 7 b. center (4, 3), tangent to the y-axis c. center (2, −1) and contains the point (4, 5) d. diameter, with endpoints (2, −5) and (4, 1)

4. Explain how you can tell by looking at the equation of the circle that the center of the circle lies on the y-axis.

5. Write the equation of a circle centered at the origin with the given radius. a. radius = 3 b. radius = 11

6. Which equation represents a circle with a diameter at endpoints (−1, 5) and (3, 7)? A. (x − 1)2 + (y − 6)2 = 5 B. (x − 2)2 + (y − 10)2 = 4 C. (x + 6)2 + (y + 1)2 = 25 D. (x + 10)2 + (y − 2)2 = 10

7. Which circle has a center of (0, –4)? A. (x − 4)2 + y2 = 4 B. (x + 1)2 + (y − 4)2 = 4 C. x2 + (y + 4)2 = 16 D. (x + 4)2 + (y − 4)2 = 16

8. Find a series of transformations that maps the circle (x − 2)2 + (y + 1)2 = 1 to the circle (x + 3)2 + y2 = 1.

9. A circle with radius 3 is centered at the origin. It is translated 2 units right and 3 units down, then dilated by a factor of 2. What is the equation for the transformed circle?

10. Identify the center and radius of the circle given by each equation. a. (x − 3)2 + y2 = 1 b. (x + 2)2 + (y + 1)2 = 16



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Lesson 32-2 11. Identify the center and radius of the circle

represented by the equation(x + 4)2 + y2 − 10y = 11.

12. Circle Q is represented by the equation (x + 1)2 + y2 + 6y = 5. Howard states that the center of the circle lies in Quadrant III. Do you agree? Justify your reasoning.

13. Circle P is represented by the equation (x − 11)2 + y2 + 10y = 25. What is the radius of the circle? A. 5 B. 10 C. 25 D. 50

14. Determine if the equation given is representative of a circle. If it is, determine the coordinates of the center of the circle. a. x + 5 − y2 − y = 5 b. x2 − 2 + y2 − 8y = 7 c. x2 + 2x + y2 − 16y = 0

15. For any equations in Item 14 that are not equations of a circle, explain how you determined your answer.

MATHEMATICAL PRACTICESLook For and Make Use of Structure

16. Given an equation of a circle, how can you determine when you need to complete the square to determine the coordinates of the center of the circle? Write an example of an equation of a circle where you would have to complete the square to determine that the circle has a radius of 10 and a center of (4, 6).




Documents

32 Equation of a Circle Directed Round and Round Lesson 32 ...(x 2+ 3)2 + y = 10 10. Determine the series of transformations that map the circle (x − 2)2 + (y + 5) 2 = 1 to the circle