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Mastering Equations of Circles in the Coordinate Plane

Learn how to write equations and graph circles in the coordinate plane, solve circle-related problems, identify centers and radii, and more. Practice converting equations and graphing circles effectively.

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Mastering Equations of Circles in the Coordinate Plane

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  1. SWBAT: Write the Equations of Circles in the Coordinate Plane 1. An ant crawls around the perimeter of a circle whose radius is 2 feet through a central angle of 72. What distance, to the nearest inch, has the ant crawled? 12 in = 1 ft

  2. SWBAT: Write the Equations of Circles in the Coordinate Plane HANDOUT • Warm Up: • In the given figure below of , and circular arc ACB and AOB is a straight line. Find the perimeter of the shaded region. • 2. Using your answer and diagram from question #1, find the area of the shaded region in terms of pi.

  3. SWBAT: Write the Equations of Circles in the Coordinate Plane • Warm Up: • In the given figure below of , and circular arc ACB and AOB is a straight line. Find the perimeter of the shaded region. • 2. Using your answer and diagram from question #1, find the area of the shaded region in terms of pi.

  4. SWBAT: Write the Equations of Circles in the Coordinate Plane 2. Isosceles triangle ABC is shown below with legs that measure 8 inches and a vertex angle of 50 . (a) Determine the area of ABC. Note that you will need to use right triangle trigonometry. Round to the nearest tenth of a square inch.  24.5 in2

  5. SWBAT: Write the Equations of Circles in the Coordinate Plane (b) Determine the area of the circular sector. Again, round to the nearest tenth of a square inch. (c) Using your answers from (a) and (b), determine the area of the shaded region.

  6. SWBAT: Write the Equations of Circles in the Coordinate Plane Objectives Write equations and graph circles in the coordinate plane. Use the equation and graph of a circle to solve problems.

  7. SWBAT: Write the Equations of Circles in the Coordinate Plane The equation of a circle is based on the Distance Formula and the fact that all points on a circle are equidistant from the center.

  8. SWBAT: Write the Equations of Circles in the Coordinate Plane

  9. SWBAT: Write the Equations of Circles in the Coordinate Plane Example 1: Writing the Equation of a Circle Write the equation of each circle. J with center J (2, 2) and radius 4 (x – h)2 + (y – k)2 = r2 Equation of a circle Substitute 2 for h, 2 for k, and 4 for r. (x – 2)2 + (y – 2)2 = 42 (x – 2)2 + (y – 2)2 = 16 Simplify.

  10. SWBAT: Write the Equations of Circles in the Coordinate Plane Practice 1: Writing the Equation of a Circle Write the equation of each circle. L with center J (-5, -6) and radius 9 (x – h)2 + (y – k)2 = r2 Equation of a circle Substitute -5 for h, -6 for k, and 9 for r. (x – (-5))2 + (y – (-6))2 = 92 (x + 5)2 + (y + 6)2 = 81 Simplify.

  11. SWBAT: Write the Equations of Circles in the Coordinate Plane Example 2: Identifying the center and radius from the equation of a circle

  12. SWBAT: Write the Equations of Circles in the Coordinate Plane Example 3: Identifying the center and radius from the graph of a circle

  13. SWBAT: Write equations and graph circles in the coordinate plane. P with center (0,-3) and passes through the point (6,5) Step 1: Calculate radius Step 2: Plug in center and radius into formula. (0, -3) (6, 5) = = 6, 8 = , (x – h)2 + (y – k)2 = r2 Substitute h = 0, k = -3, and r2 = 100 x2 + (y + 3)2 = 100

  14. SWBAT: Write equations and graph circles in the coordinate plane. Concept 5: Writing the equation of a circle given two endpoints on the diameter Writing the equation of  K that passes through endpoints A(5, 4) and B(1, –8). Center = = (3, -2) = (3, -2) (5, 4) = = 2, 6 (x – 3)2 + (y + 2)2 = 40

  15. SWBAT: Write equations and graph circles in the coordinate plane. Practice #2 Write the equation of circleQ that passes through endpoints (2, 3) and (2, –1) Center = = (2, 1) = (2, 1) (2, 3) = = 0, 2 (x – 2)2 + (y - 1)2 = 4

  16. SWBAT: Write equations and graph circles in the coordinate plane. Example 6: To go from General Form to Standard Form (center-radius form) you have to complete the square twice: + + 12 = 0 + __ + + __ = -12 + __ + __ x2 + 4x + 4 + y2 - 6y + 9 = -12 + 4 + 9 (x + 2)2 + (y - 3)2 = 1 Center = (-2, 3) r =

  17. SWBAT: Write the Equations of Circles in the Coordinate Plane x2 + 6x + y2 – 8y = -24 x2 + 6x + __ + y2 – 8y + __ = -24 + __ + __ x2 + 6x + 9 + y2 – 8y + 16 = -24 + 9 + 16 (x + 3)2 + (y – 4)2 = 1 Center = (-3, 4) r = 1

  18. SWBAT: Write the Equations of Circles in the Coordinate Plane If you are given the equation of a circle, you can graph the circle by identifying its center and radius.

  19. SWBAT: Write the Equations of Circles in the Coordinate Plane Example 7: Graphing a Circle Graph x2 + y2 = 16.

  20. SWBAT: Write the Equations of Circles in the Coordinate Plane Practice: Graphing a Circle Graph (x + 5)2 + (y - 2)2 = 4.

  21. SWBAT: Write the Equations of Circles in the Coordinate Plane

  22. SWBAT: Write the Equations of Circles in the Coordinate Plane

  23. SWBAT: Write the Equations of Circles in the Coordinate Plane

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