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Equations and Graphs of Circles: Center, Radius, and Point

Learn how to write equations of circles using the center and radius or a point. Practice graphing circles on the coordinate plane and use three points to write an equation.

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Equations and Graphs of Circles: Center, Radius, and Point

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 10–7) NGSSS Then/Now New Vocabulary Key Concept: Standard Form, Equation of a Circle Example 1: Write an Equation Using the Center and Radius Example 2: Write an Equation Using the Center and a Point Example 3: Graph a Circle Example 4: Real-World Example: Use Three Points to Write an Equation Lesson Menu

  3. A B C D Find x. A. 1 B. 2 C. 3 D. 4 5-Minute Check 1

  4. A B C D Find x. A. 1 B. 2 C. 3 D. 4 5-Minute Check 2

  5. A B C D Find x. A. 2 B. 4 C. 6 D. 8 5-Minute Check 3

  6. A B C D Find x. A. 10 B. 9 C. 8 D. 7 5-Minute Check 4

  7. A B C D A.14 B. C. D. Find x in the figure. 5-Minute Check 5

  8. MA.912.G.6.6Given the center and the radius, find the equation of a circle in the coordinate plane or given the equation of a circle in center-radius form, state the center and the radius of the circle. MA.912.G.6.7 Given the equation of a circle in center-radius form or given the center and the radius of a circle, sketch the graph of the circle. NGSSS

  9. You wrote equations of lines using information about their graphs. (Lesson 3–4) • Write the equation of a circle. • Graph a circle on the coordinate plane. Then/Now

  10. compound locus Vocabulary

  11. Concept

  12. Write an Equation Using the Center and Radius A. Write the equation of the circle with a center at (3, –3) and a radius of 6. (x – h)2 + (y – k)2 = r2 Equation of circle (x – 3)2 + (y – (–3))2 = 62 Substitution (x – 3)2 + (y + 3)2 = 36 Simplify. Answer:(x – 3)2 + (y + 3)2 = 36 Example 1

  13. Write an Equation Using the Center and Radius B. Write the equation of the circle graphed to the right. The center is at (1, 3) and the radius is 2. (x – h)2 + (y – k)2 = r2 Equation of circle (x – 1)2 + (y – 3)2 = 22 Substitution (x – 1)2 + (y – 3)2 = 4 Simplify. Answer:(x – 1)2 + (y – 3)2 = 4 Example 1

  14. A B C D A. Write the equation of the circle with a center at (2, –4) and a radius of 4. A.(x – 2)2 + (y + 4)2 = 4 B.(x + 2)2 + (y – 4)2 = 4 C.(x – 2)2 + (y + 4)2 = 16 D.(x + 2)2 + (y – 4)2 = 16 Example 1

  15. A B C D B. Write the equation of the circle graphed to the right. A.x2 + (y + 3)2 = 3 B.x2 + (y – 3)2 = 3 C.x2 + (y + 3)2 = 9 D.x2 + (y – 3)2 = 9 Example 1

  16. Write an Equation Using the Center and a Point A. Write the equation of the circle that has its center at (–3, –2) and passes through (1, –2). Step 1Find the distance between the points to determine the radius. Distance Formula (x1, y1) = (–3, –2) and(x2, y2) = (1, –2) Simplify. Example 2

  17. Write an Equation Using the Center and a Point Step 2Write the equation using h = –3, k = –2, andr = 4. (x – h)2 + (y – k)2 = r2 Equation of circle (x – (–3))2 + (y – (–2))2 = 42 Substitution (x + 3)2 + (y + 2)2 = 16 Simplify. Answer:(x + 3)2 + (y + 2)2 = 16 Example 2

  18. The center is at (–2, –4) and the radius is Write an Equation Using the Center and a Point B. Write the equation of the circle graphed to the right. Step 1Identify the center of the circle and find the distance of the radius. Example 2

  19. Step 2Write the equation using h = –2, k = –4, and (x – (–2))2 + (y – (–4))2 = Substitution Write an Equation Using the Center and a Point (x – h)2 + (y – k)2 = r2 Equation of circle (x + 2)2 + (y + 4)2 = 17 Simplify. Answer:(x + 2)2 + (y + 4)2 = 17 Example 2

  20. A B C D A. Write the equation of the circle that has its center at (–1, 0) and passes through (3, 0). A. (x + 1)2 + y2 = 16 B. (x – 1)2 + y2 = 16 C. (x + 1)2 + y2 = 4 D. (x – 1)2 + y2 = 16 Example 2

  21. A B C D A. B. C. D. B. Write the equation of the circle graphed to the right. Example 2

  22. Graph a Circle The equation of a circle is (x – 2)2 + (y + 3)2 = 4. State the center and the radius. Then graph the equation. Compare each expression in the equation to the standard form. r2 = 4, so r = 2. Example 3

  23. Graph a Circle Answer: The center is at (2, –3), and the radius is 2. Example 3

  24. A B C D A.B. C.D. Which of the following is the graph of x2 + (y – 5)2 = 25? Example 3

  25. Use Three Points to Write an Equation ELECTRICITY Strategically located substations are extremely important in the transmission and distribution of a power company’s electric supply. Suppose three substations are modeled by the points D(3, 6), E(–1, 0), and F(3, –4). Determine the location of a town equidistant from all three substations, and write an equation for the circle. Understand You are given three points that lie on a circle. Plan Graph ΔDEF. Construct the perpendicular bisectors of two sides to locate the center, which is the location of the tower. Find the length of a radius. Use the center and radius to write an equation. Example 4

  26. Use Three Points to Write an Equation Solve Graph ΔDEF and construct the perpendicular bisectors of two sides. Example 4

  27. Use Three Points to Write an Equation The center, C, appears to be at (4, 1). This is the location of the tower. Find r by using the Distance Formula with the center and any of the three points. Write an equation. Example 4

  28. Use Three Points to Write an Equation Answer: The location of a town equidistant from all three substations is at (4,1). The equation for the circle is (x – 4)2 + (y – 1)2 = 26. Check You can verify the location of the center by finding the equations of the two bisectors and solving a system of equations. You can verify the radius by finding the distance between the center and another of the three points on the circle. Example 4

  29. A B C D AMUSEMENT PARKS The designer of an amusement park wants to place a food court equidistant from the roller coaster located at (4, 1), the Ferris wheel located at (0, 1), and the boat ride located at (4, –3). Determine the location for the food court. A. (3, 0) B. (0, 0) C. (2, –1) D. (1, 0) Example 4

  30. End of the Lesson

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