The Equation of a Circle What to KNOW What to KNOW

Find out how much the students have learned about the different mathematics concepts previously studied and their skills in performing mathematical operations. Checking these will facilitate teaching and students’

understanding of the equation of a circle. Tell them that as they go through this lesson, they have to think of this important question: “How does the equation of a circle facilitate finding solutions to real-life problems and making decisions?”

Two of the essential mathematics concepts needed by the students in understanding the equation of a circle are the perfect square trinomial and the square of a binomial. Activity 1 of this lesson will provide them opportunity to recall these concepts. In this activity, the students will determine the number that must be added to a given expression to make it a perfect square trinomial and then express the result as a square of a binomial. They should be able to explain how they came up with the perfect square trinomial and the square of a binomial.

Emphasize to the students that the process they have done in producing a perfect square trinomial is also referred to as completing the square.

Activity 1: Make It Perfect!

Answer Key 1. 4;



x2



²

2. 25;



t5



² 3. 49;



r 7



² 4. 121;



r ¹¹



² 5. 324;



x18



²

a. Add the square of one-half the coefficient of the linear term.

b. Factor the perfect square trinomial.

c. Use the distributive property of multiplication or FOIL Method.

222

Provide the students opportunity to develop their understanding of the equation of a circle. Ask them to perform Activity 2. In this activity, the students will be presented with a situation involving the equation of a circle. Let them find the distance of the plane from the air traffic controller given the coordinates of the point where it is located and the y-coordinate of the position of the plane at a particular instance if its x-coordinate is given. Furthermore, ask them to describe the path of the plane as it goes around the airport. Challenge them to determine the equation that would define the path of the plane. Let them realize that the distance formula is related to the equation defining the plane’s path around the airport.

Activity 2: Is there a traffic in the air?

Answers Key

Provide the students opportunity to come up with an equation that can be used in finding the radius of a circle. Ask them to perform Activity 3. In this activity, the students should be able to realize that the Distance Formula can be used in finding the radius of a circle. And that the distance of a point from the center of a circle is also the radius of the circle.

Answer Key

3. No. It is not possible for the plane to be at a point whosexcoordinate is 60 because its distance from the air traffic controller would be

Activity 3: How far am I from my point of rotation?

A.

Answer Key 1. 8 units

2. Yes, the circle will pass through

 

0 , ,8



8,0



, and



0,8



because the distance from these points to the center of the circle is 8 units.

3. No, because the distance from point M



4,6



to the center of the circle is less than 8 units.

No, because the distance from point N



9,2



to the center of the circle is more than 8 units.

4. 8 units; 80 = 8

5. If a point is on the circle, its distance from the center is equal to the radius.

6. Since the distance d of a point from the center of the circle is 2 y2

x

d   and is equal to the radius r, then r  x2 y 2or 2

2

2 y r

x   .

y

x

224 B.

Before proceeding to the next activities, let the students give a brief summary of the activities they have done. Provide them with an opportunity to relate or connect their responses in the activities given to their new lesson, equation of a circle. Let the students read and understand some important notes on equation of a circle. Tell them to study carefully the examples given.

What to PROCESS

Let the students use the mathematical ideas they have learned about the equation of a circle and the examples presented in the preceding section to perform the succeeding activities.

Answer Key

1. ⁶¹ units or approximately 7.81 units 2. Yes, the circle will pass through



2,7



,

 

8 , and ,7



3,4



because the distance from each of these points to the center of the circle is ⁶¹ units or approximately 7.81 units.

3. No, because the distance from point M



7,6



to the center of the circle is more than 7.81 units.

4. ⁶¹ units or approximately 7.81 units.

Note: Evaluate students’ explanations.

5. If the center of the circle is not at the origin, its radius can be determined by using the distance formula,



^x₂ ^x₁

 

^{2 y}₂ ^y₁



²

d     . Since the distance of the point from the center of the circle is equal to the radius r, then



^x₂ ^x₁

 

^{2 y}₂ ^y₁



²

r     or



x₂x₁

 

²  y₂y₁



²r². If

 

x,y

P is a point on the circle and C

 

h,k is the center, then



^x₂^x₁

 

^{2 y}₂ ^y₁



^{2 r2} becomes



^xh

 

^{2  y k}



^{2 r2}. y

x

In Activity 4, the students will determine the center and the radius of each circle, given its equation. Then, the students will be asked to graph the circle. Ask them to explain how they determined the center and the radius of the circle.

Furthermore, tell them to explain how to graph a circle given its equation in different forms. Strengthen students’ understanding of the graphs of circles through the use of available mathematics freeware like Geogebra.

Activity 4: Always Start at This Point!

Answer Key

1. Center:

 

_{0 ,0} 3. Center:

 

_{0 ,0}

Radius: 7 units Radius: 10 units

2. Center:

 

5 ,6 4. Center:



7,1



Radius: 9 units Radius: 7 units y

x

y

x

y

x

y

x

226

Ask the students to perform Activity 5. This time, the students will write the equation of a circle given the center and the radius. Ask them to explain how to determine the equation of a circle whether or not the center is the origin.

Activity 5: What defines me?

Answer Key 1. x²y² 144

2.



x2

 

² y6



² 81 3.



x7

 

² y 2



² 225 4.



x4

 

² y5



² 50 5.



x10

 

² y8



² 27

Answer Key

5. Center:



4,3



6. Center:



5,8



Radius: 8 units Radius: 11 units

a. Note: Evaluate students’ responses.

b. Determine first the center and the radius of the circle defined by the equation, then graph.

If the given equation is in the formx²y² r², the center is at the origin and the radius of the circle is r.

If the given equation is in the form



xh

 

² y k



² r², the center is at

 

h,k and the radius of the circle is r.

If the given equation is in the formx²y²DxEyF 0, transform it into the form



xh

 

² y k



² r². The center is at

 

h,k and the radius of the circle is r.

a. Write the equation in the

formx² y² r²where the origin is the center and r is the radius of the circle.

Write the equation in the

form



xh

 

² yk



² r²where

 

h,k is the center and r is the radius of the circle.

b. No, because the two circles have different radii.

y

x

y

x

Activities 6 and 7 provide students opportunities to write equations of circles from center-radius form or standard form to general form and vice-versa.

At this point, ask them to explain how to transform the equation of a circle from one form to another form and discuss the mathematics concepts or principles applied. Furthermore, challenge them to find a shorter way of transforming equation of a circle from general form to standard form and vice-versa.

Activity 6: Turn Me into a General!

Answer Key

Note: Evaluate students’ explanations.

Activity 7: Don’t Treat this as a Demotion!

Answer Key property of equality and factoring.

228

b. Completing the square, Addition Property of Equality, Square of a Binomial

c. Using the values of D, E, and F in the general equation of a circle, 2 0

2y DxEyF

x , to find the center (h,k) and radius r. The GeoGebra freeware can also be used for verification.

What to REFLECT on and UNDERSTAND:

Ask the students to have a closer look at some aspects of the equation of a circle. Provide them with opportunities to think deeply and test further their understanding of the equation of a circle by doing Activities 8 and 9. Give more focus on the real-life applications of the equation of a circle.

Activity 8: A Circle? Why not?

Activity 9: Find Out More!

Answer Key

1. No. x2 y22x8y 260 can be written as



^x1

 

^{2 y 4}



^29. Notice that -9 cannot be expressed as a square of another number.

2. Yes. x2y294x10y can be written as



x2

 

2 y5



2 20_. 3. No. x2 y26x8y 32 is not an equation of a circle. Its graph is not

also a circle.

4. No. x2y28x14y 650 is merely a point. The radius must be greater than 0 for a circle to exist.

Answer Key

1.



x3

 

² y8



² 81

2.



x10

 

² y 7



² 36or



x10

 

² y5



² 36 3. 3x5y 7

4.



x5

 

² y5



² 13 5. a.



x3

 

² y4



² 100

b. Yes, because point

 

11,6 is still within the critical area.

c. Follow the advice of PDRRMC.

d. (Evaluate students’ responses/explanations.)

Before the students move to the next section of this lesson, give a short test (formative test) to find out how well they understood the lesson. Ask them also to write a journal about their understanding of the equation of a circle. Refer to the Assessment Map.

What to TRANSFER

Give the students opportunities to demonstrate their understanding of the equation of a circle by doing a practical task. Let them perform Activity 10. You can ask the students to work individually or in a group.

In Activity 10, the students will paste some small pictures of objects on grid paper and position them at different coordinates. Then, the students will draw circles that contain these pictures. Using the pictures and the circles drawn on the grid, they will formulate problems involving the equation of the circle, and then solve them.

Activity 10: Let This be a Part of My Scrapbook!

Answer Key

Evaluate students’ answers. You may use the rubric.

Answer Key

6. a. Wise Tower -



x5

 

² y3



² 81 Global Tower -



x3

 

² y6



² 16 Star Tower -



x12

 

² y 3



² 36 b.



12,2



- Star Tower



6,7



- Wise Tower

 

2,8 - Global Tower

 

,13 - Wise and Global Tower

c. Many possible answers. Evaluate students’ responses.

230 Summary/Synthesis/Generalization:

This lesson was about the equation of circles. The lesson provided the students with opportunities to illustrate the center-radius form of the equation of a circle, determine the center and the radius of a circle given its equation and vice versa, write the equation of a circle from standard form to general form and vice-versa, graph circles on the coordinate plane, and solve problems involving the equation of circles. Moreover, they were given the opportunity to formulate and solve real-life problems involving the equation of a circle through the practical task performed. Their understanding of this lesson and other previously learned mathematics concepts and principles will facilitate their learning of other related mathematics concepts.

SUMMATIVE TEST

In document Math10 Tg u2 (Page 160-170)