4.1 Circles. Explore Deriving the Standard-Form Equation

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1 COMMON CORE r Locker LESSON Circles.1 Name Class Date.1 Circles Common Core Math Standards The student is epected to: COMMON CORE A-CED.A.3 Represent constraints b equations or inequalities,... and interpret solutions as viable or nonviable options in a modeling contet. Also A-CED.A., G-GPE.A.1, G-GPE.B. Mathematical Practices COMMON CORE MP.7 Using Structure Language Objective Work with a partner to match graphs of circles to their equations in standard form. ENGAGE Essential Question: What is the standard form for the equation of a circle, and what does the standard form tell ou about the circle? Possible answer: The standard form for the equation of a circle is ( - h) + ( - k) = r, which tells ou that the center is (h, k) and the radius is r. PREVIEW: LESSON PERFORMANCE TASK View the online Engage. Discuss the photo and the generall circular nature of radio-signal reception strength. Then preview the Lesson Performance Task. Essential Question: What is the standard form for the equation of a circle, and what does the standard form tell ou about the circle? Eplore Deriving the Standard-Form Equation of a Circle Recall that a circle is the set of points in a plane that are a fied distance, called the radius, from a given point, called the center. A B C The coordinate plane shows a circle with center C(h, k) and radius r. P(, ) is an arbitrar point on the circle but is not directl above or below or to the left or right of C. A(, k) is a point with the same -coordinate as P and the same -coordinate as C. Eplain wh CAP is a right triangle. Since point A has the same -coordinate as point P, segment PA is a vertical segment. Since point A has the same -coordinate as point C, segment CA is a horizontal segment. This means that segments PA and CA are perpendicular, which means that CAP is a right angle and CAP is a right triangle. Identif the lengths of the sides of CAP. Remember that point P is arbitrar, so ou cannot rel upon the diagram to know whether the -coordinate of P is greater than or less than h or whether the -coordinate of P is greater than or less than k, so ou must use absolute value for the lengths of the legs of CAP. Also, remember that the length of the hpotenuse of CAP is just the radius of the circle. - h The length of segment AC is. - k The length of segment AP is. The length of segment CP is r. Appl the Pthagorean Theorem to CAP to obtain an equation of the circle. ( - h ) + ( - k ) = r r C (h, k) A (, k) Resource Locker P (, ) Module 159 Lesson 1 Name Class Date.1 Circles Essential Question: What is the standard form for the equation of a circle, and what does the standard form tell ou about the circle? A-CED.A.3 Represent constraints b equation or inequalities,... and interpret solutions as viable or nonviable options in a modeling contet. Also A-CED.A., G-GPE.A.1, G-GPE.B. Eplore Deriving the Standard-Form Equation of a Circle Recall that a circle is the set of points in a plane that are a fied distance, called the radius, from a given point, called the center. The coordinate plane shows a circle with center C(h, k) and radius r. P(, ) is an arbitrar point on the circle but is not directl above or below or to the left or right of C. A(, k) is a point with the same -coordinate as P and the same -coordinate as C. Eplain wh CAP is a right triangle. Since point A has the same -coordinate as point P, segment PA is a vertical segment. Since point A has the same -coordinate as point C, segment CA is a horizontal segment. This means that segments PA and CA are perpendicular, which means that CAP is a right angle and CAP is a right triangle. Identif the lengths of the sides of CAP. Remember that point P is arbitrar, so ou cannot rel upon the diagram to know whether the -coordinate of P is greater than or less than h or whether the -coordinate of P is greater than or less than k, so ou must use absolute value for the lengths of the legs of CAP. Also, remember that the length of the hpotenuse of CAP is just the radius of the circle. The length of segment AC is. The length of segment AP is. The length of segment CP is. Appl the Pthagorean Theorem to CAP to obtain an equation of the circle. = ( - ) + ( - ) - h - k r h k r Resource P (, ) C (h, k) A (, k) Module 159 Lesson 1 HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition. 159 Lesson.1

2 Reflect 1. Discussion Wh isn t absolute value used in the equation of the circle? Since squaring removes an negative signs just as absolute value does, there s no need to take absolute value before squaring.. Discussion Wh does the equation of the circle also appl to the cases in which P has the same -coordinate as C or the same -coordinate as C so that CAP doesn t eist? If P has the same -coordinate as C, then P s -coordinate must be either k + r or k - r. So, ( - h) + ( - k) = (h - h) + ( (k ± r) - k) = 0 + (±r) = r, and the equation of the circle is still satisfied. Similarl, if P has the same -coordinate as C, then P s -coordinate must be either h + r or h - r. So, ( - h) + ( - k) = ((h ± r) - h) + (k - k) = (±r) + 0 = r, and the equation of the circle is still satisfied. EXPLORE Deriving the Standard-Form Equation of a Circle INTEGRATE TECHNOLOGY Students have the option of completing the Eplore activit either in the book or online. Eplain 1 Writing the Equation of a Circle The standard-form equation of a circle with center C(h, k) and radius r is ( - h) + ( - k) = r. If ou solve this equation for r, ou obtain the equation r = ( -h) + ( - k), which gives ou a means for finding the radius of a circle when the center and a point P(, ) on the circle are known. Eample 1 Write the equation of the circle. The circle with center C(-3, ) and radius r = Substitute -3 for h, for k, and for r into the general equation and simplif. ( - (-3) ) + ( - ) = ( + 3) + ( - ) = 16 The circle with center C(, -3) and containing the point P(, 5) Step 1 Find the radius. r = CP = ( - ()) 5 = ( 6 ) + ( 8 ) = = _ 100 = 10 Step Write the equation of the circle. ( - () ) + ( - (-3) ) ( + ) + ( + 3) = + ( - (-3)) = Module 160 Lesson 1 Wh does point A have coordinates (, k)? It is below P (, ), which has -coordinate, and on the same horizontal line as C (h, k), which has -coordinate k. Wh is it necessar to use absolute value signs when representing the length of the legs of the right triangle? Since P could be an point on the circle, absolute value signs are used to make sure the length of each leg is a positive number. EXPLAIN 1 Writing the Equation of a Circle AVOID COMMON ERRORS Some students ma forget to square the radius when writing the equation. Others ma take its square root. Help students to avoid making these errors b having them write r above the place in the equation where the need to write the square of the radius. PROFESSIONAL DEVELOPMENT Math Background The equation of a circle is based on the fact that all of the points on the circle are a fied distance from a given point. This distance can be found using the Pthagorean Theorem. In the derivation, the fied distance, the radius, is represented b the hpotenuse of a right triangle that has one verte at the center of the circle, and one on the circle itself. Appling the Pthagorean Theorem produces the equation of the circle, ( - h) + ( - k) = r. Note that taking the square root of each side of this equation produces the equation ( - h) + ( - k) = r. This equation shows that the radius is the distance between the two points. Do ou need to be given the coordinates of the center to write an equation of a circle? If not, what other information could ou use to find the center? Eplain. No. If ou know the endpoints of a diameter of the circle, ou can find the coordinates of the center using the Midpoint Formula. Circles 160

3 INTEGRATE MATHEMATICAL Focus on Math Connections MP.1 Discuss with students how the graph of the equation ( - h) + ( - k) = r is a transformation of the graph of + = r. Have students describe the transformation and compare the two graphs. EXPLAIN Rewriting an Equation of a Circle to Graph the Circle How do ou know what number to add to make perfect square trinomials when converting to standard form? Take half of the coefficient of the term, and square it. Then do the same with the coefficient of the term. Once the equation is in standard form, how do ou find the diameter of the circle? The diameter is times the square root of the number that represents r. Your Turn Write the equation of the circle. 3. The circle with center C(1, ) and radius r = ( - 1) + ( - () ) =. The circle with center C(-, 5) and containing the point P(-, -1) Eplain Rewriting an Equation of a Circle to Graph the Circle Epanding the standard-form equation ( - h) + ( - k) = r results in a general second-degree equation in two variables having the form + + c + d + e = 0. In order to graph such an equation or an even more general equation of the form a + a + c + d + e = 0. ou must complete the square on both and to put the equation in standard form and identif the circle s center and radius. Eample ( - 1) + ( + ) = Because points C and P have the same -coordinate, the radius of the circle is just the absolute value of the difference of their -coordinates, so r = 5 - (-1) = 6 = 6. ( - (-) ) + ( - 5) = 6 ( + ) + ( - 5) = 36 Graph the circle after writing the equation in standard form = 0 Write the equation = 0 Prepare to complete the square on and. Complete both squares. ( ) + ( ) = Factor and simplif. ( - 5) + ( + 3) = The center of the circle is C(5, -3), and the radius is r = _ =. Graph the circle. ( ) + ( ) = Module 161 Lesson 1 COLLABORATIVE LEARNING Peer-to-Peer Activit Have students work in pairs. Instruct each student in each pair to write an equation of a circle in standard form. Have them graph the circles, keeping their graphs hidden from their partners. Have them also convert their equations to the general form a + b + c + d + e = 0 b epanding and combining like terms. Instruct students to echange the general forms of their equations, write each other s equations in standard form, and draw the graph. Have students compare their work. 161 Lesson.1

4 B = 0 Write the equation = 0 Factor from the terms ( + ) + ( - ) + 11 = 0 and the terms. Prepare to complete ( + + ) + ( - + ) = ( ) + ( ) the square on and. Complete both ( ) + ( - + ) = ( 1 ) + ( ) squares. Factor and simplif. ( + 1 ) + ( - ) = Divide both sides b. ( + 1 ) + ( - ) = The center is C (, ), and the radius is r = _ =. Graph the circle _ 3_ 9 9_ AVOID COMMON ERRORS When adding the numbers to the constant term to maintain the equalit, students ma forget to multipl the number added to complete each square b the leading coefficient that was factored from the variable terms. Help them to avoid this error b encouraging them to circle the leading coefficients that have been factored out in the process of completing the square. INTEGRATE MATHEMATICAL Focus on Critical Thinking MP.3 Have students compare how the completing-the-square process is used to write quadratic functions in verte form with how it is used to write a circle in standard form. Have them describe both the similarities and the differences. Your Turn Graph the circle after writing the equation in standard form = = 0 ( + ) + ( + 6) = ( + + ) + ( ) = ( + ) + ( + 3) = 9 The center is C(-, -3), and the radius is r = 9 = = = 0 9 ( - 6) + 9 ( - 8) = ( ) + 9 ( ) = (9) + 9 (16) ( - 3) + 9 ( - ) = 16 16_ ( - 3) + ( - ) = 9 - The center is C (3, ), and the radius is r = _ 16 _ 9 = 3. Module 16 Lesson DIFFERENTIATE INSTRUCTION Kinesthetic Eperience Prepare a length of string with a marker attached at one end. Displa a coordinate plane. With a tack, tape, or some other means, attach the non-marker end of the string to an arbitrar point on the plane and draw a circle. Ask students to eplain how the could use a point on the circle and the center to find the length of the string. Help them to see that the can construct a right triangle whose hpotenuse is the length of the string, and appl the Pthagorean Theorem to determine the length of the string. Circles 16

5 EXPLAIN 3 Solving a Real-World Problem Involving a Circle If ou know the equation of a circle, how can ou determine whether a given point lies inside, outside, or on the circle? You can substitute the coordinates of the point for and in the equation of the circle, and see whether the value of ( - h) + ( - k) is less than, greater than, or equal to the value of r. If it is less than r, the point lies inside the circle; if greater than, the point lies outside the circle; and if equal to, the point lies on the circle. INTEGRATE MATHEMATICAL Focus on Reasoning MP. Ask students to eplain wh points that are inside the circle satisf the inequalit ( - h) + ( - k) < r. Students should recognize that a point inside the circle would lie on a circle whose radius would be shorter than r, and whose equation would be ( - h) + ( - k) = r 1, with r 1 < r. Thus, r 1 < r and ( - h) + ( - k) < r. Image Credits: Jose Luis Pelaez/Corbis Eplain 3 Solving a Real-World Problem Involving a Circle A circle in a coordinate plane divides the plane into two regions: points inside the circle and points outside the circle. Points inside the circle satisf the inequalit ( - h) + ( - k) < r, while points outside the circle satisf the inequalit ( - h) + ( - k) > r. Eample 3 Write an inequalit representing the given situation, and draw a circle to solve the problem. The table lists the locations of the homes of five friends along with the locations of their favorite pizza restaurant and the school the attend. The friends are deciding where to have a pizza part based on the fact that the restaurant offers free deliver to locations within a 3-mile radius of the restaurant. At which homes should the friends hold their pizza part to get free deliver? Place Location Alonzo s home A(3, ) Barbara s home B(, ) Constance s home C(-, 3) Dion s home D(0, -1) Eli s home E(1, ) Pizza restaurant (-1, 1) School (1, -) Write the equation of the circle with center (-1, 1) and radius 3. ( - (-1)) + ( - 1) = 3, or ( + 1) + ( - 1) = 9 The inequalit ( + 1) + ( - 1) < 9 represents the situation. Plot the points from the table and graph the circle. The points inside the circle satisf the inequalit. So, the friends should hold their pizza part at either Constance s home or Dion s home to get free deliver. B C A Restaurant - 0 D - School E In order for a student to ride the bus to school, the student must live more than miles from the school. Which of the five friends are eligible to ride the bus? Write the equation of the circle with center ( 1, - ) and radius. ( - 1 ) + ( - ( - )) = ( - 1 ) + ( + ) = The inequalit ( - 1 ) + ( + ) > represents the situation. Use the coordinate grid in Part A to graph the circle. The points outside the circle satisf the inequalit. So, Alonzo, Barbara, and Constance are eligible to ride the bus. Module 163 Lesson 1 LANGUAGE SUPPORT Communicate Math Have students work in pairs. Provide each pair of students with different graphs of circles and, on separate note cards or sheets of paper, the equations for those graphs. The first student chooses a graph and decides which equation goes with it, then eplains wh the are a match. The second student repeats the procedure using another graph and equation. 163 Lesson.1

6 Reflect 7. For Part B, how do ou know that point E isn t outside the circle? The coordinates of point E are (1, ). Substituting 1 for and for in ( - 1) + ( + ) gives (1-1) + ( + ) = 0 + (-) =, so the coordinates of E satisf the equation of the circle, which means that E is on the circle and not outside it. Your Turn Write an inequalit representing the given situation, and draw a circle to solve the problem. 8. Sasha delivers newspapers to subscribers that live within a -block radius of her house. Sasha s house is located at point (0, -1). Points A, B, C, D, and E represent the houses of some of the subscribers to the newspaper. To which houses does Sasha deliver newspapers? ( - 0) + ( - (-1)) = Elaborate + ( + 1) = 16 The inequalit + ( + 1) < 16 represents the situation. The points inside the circle satisf the inequalit + ( + 1) <16. So, Sasha delivers to the houses located at points B, D, and E. C A B - 0 Sasha - D E 9. Describe the process for deriving the equation of a circle given the coordinates of its center and its radius. First, choose an arbitrar point P on the circle. Net, find a third point A that forms a right triangle with points C and P. Then, use the coordinates of the three points to find the lengths of segments CA and PA. (The length of segment CP is the circle s radius.) Finall, use the Pthagorean Theorem to write an equation of the circle. ELABORATE How is the equation of a circle related to the equation a + b = c from the Pthagorean Theorem? The radius is c, and the lengths of the legs of the right triangle that has the radius as its hpotenuse are a and b. INTEGRATE MATHEMATICAL Focus on Reasoning MP. Discuss with students wh, in the equation a + b + c + d + e = 0, a and b must be equal for the equation to be that of a circle. Focus their attention on the steps involved in converting the equation to the standard form of a circle, and on the roles of a and b in the conversion. 10. What must ou do with the equation a + a + c + d + e = 0 in order to graph it? Complete the square on and to write the equation in standard form. From the standard-form equation ou can then identif the circle s center and radius, which ou can then use to graph the circle. 11. What do the inequalities ( - h) + ( - k) < r and ( - h) + ( - k) > r represent? The inequalit ( - h) + ( - k) < r represents points inside the circle with equation ( - h) + ( - k) = r, and the inequalit ( - h) + ( - k) > r represents points outside the circle. 1. Essential Question Check-In What information must ou know or determine in order to write an equation of a circle in standard form? You must know the center of the circle and its radius to write an equation of the circle in standard form. If onl the center and a point on the circle are known, ou can determine the radius from those two points. SUMMARIZE THE LESSON How can ou write the equation of a circle? You can use the coordinates of the center for h and k, and the radius for r, in the equation ( - h) + ( - k) = r. Module 16 Lesson 1 Circles 16

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