1. Holt McDougal Geometry 12-7 Circles in the Coordinate Plane 12-7 Circles in the Coordinate Plane Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz Holt McDougal Geometry

2. 12-7 Circles in the Coordinate Plane Warm Up Use the Distance Formula to find the distance, to the nearest tenth, between each pair of points. 1. A(6, 2) and D(–3, –2) 2. C(4, 5) and D(0, 2) 3. V(8, 1) and W(3, 6) 9.8 5 7.1 4. Fill in the table of values for the equation y = x – 14.

3. Holt McDougal Geometry 12-7 Circles in the Coordinate Plane Write equations and graph circles in the coordinate plane. Use the equation and graph of a circle to solve problems. Objectives

4. Holt McDougal Geometry 12-7 Circles in the Coordinate Plane

5. Holt McDougal Geometry 12-7 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.

6. Holt McDougal Geometry 12-7 Circles in the Coordinate Plane Standard Equation of a circle Center: Radius: Equation: (h,k) r

7. Holt McDougal Geometry 12-7 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 = r 2 (x – 2) 2 + (y – 2) 2 = 4 2 (x – 2) 2 + (y – 2) 2 = 16 Equation of a circle Substitute 2 for h, 2 for k, and 4 for r. Simplify.

8. Holt McDougal Geometry 12-7 Circles in the Coordinate Plane Example 2: Writing the Equation of a Circle Write the equation of each circle. circle with center P(-1,4) and a radius of units. (x – h) 2 + (y – k) 2 = r 2 (x – -1) 2 + (y – 4) 2 = 2 (x + 1) 2 + (y – 4) 2 = 15 Equation of a circle Substitute 2 for h, 2 for k, and 4 for r. Simplify.

9. Holt McDougal Geometry 12-7 Circles in the Coordinate Plane Example 1B: Writing the Equation of a Circle Write the equation of each circle.  K that passes through J(6, 4) and has center K(1, –8) Distance formula. Simplify. (x – 1) 2 + (y – (–8)) 2 = 13 2 (x – 1) 2 + (y + 8) 2 = 169 Substitute 1 for h, –8 for k, and 13 for r. Simplify.

10. Holt McDougal Geometry 12-7 Circles in the Coordinate Plane Example 4 Write the equation of each circle.  P with center P(0, –3) and radius 8 (x – h) 2 + (y – k) 2 = r 2 (x – 0) 2 + (y – (–3)) 2 = 8 2 x 2 + (y + 3) 2 = 64 Equation of a circle Substitute 0 for h, –3 for k, and 8 for r. Simplify.

11. Holt McDougal Geometry 12-7 Circles in the Coordinate Plane Check It Out! Example 1b Write the equation of each circle.  Q that passes through (2, 3) and has center Q(2, –1) Distance formula. Simplify. (x – 2) 2 + (y – (–1)) 2 = 4 2 (x – 2) 2 + (y + 1) 2 = 16 Substitute 2 for h, –1 for k, and 4 for r. Simplify.

12. Holt McDougal Geometry 12-7 Circles in the Coordinate Plane Example 6: Determine the coordinates of the center and the measure of the radius. +121 Center: (-4,0) Radius: 11

13. Holt McDougal Geometry 12-7 Circles in the Coordinate Plane Example 7: Write the equation that has a diameter whose endpoints are at (2,7) and (-6,15) First find the center: Next find the radius:

14. Holt McDougal Geometry 12-7 Circles in the Coordinate Plane Next write the equation: (x + 2) 2 + (y - 11) 2 = 32 (x – - 2) 2 + (y – 11) 2 = 2

15. Holt McDougal Geometry 12-7 Circles in the Coordinate Plane If you are given the equation of a circle, you can graph the circle by making a table or by identifying its center and radius.

16. Holt McDougal Geometry 12-7 Circles in the Coordinate Plane Example 8: Graphing a Circle Graph (x – 3) 2 + (y + 4) 2 = 9. The equation of the given circle can be written as (x – 3) 2 + (y – (– 4)) 2 = 3 2. So h = 3, k = –4, and r = 3. The center is (3, –4) and the radius is 3. Plot the point (3, –4). Then graph a circle having this center and radius 3. (3, –4)

17. Holt McDougal Geometry 12-7 Circles in the Coordinate Plane Example 9 Graph (x – 3) 2 + (y + 2) 2 = 4. The equation of the given circle can be written as (x – 3) 2 + (y – (– 2)) 2 = 2 2. So h = 3, k = –2, and r = 2. The center is (3, –2) and the radius is 2. Plot the point (3, –2). Then graph a circle having this center and radius 2. (3, –2)

18. Holt McDougal Geometry 12-7 Circles in the Coordinate Plane Lesson Quiz: Part II Graph each equation. 3. x 2 + y 2 = 4 4. (x – 2) 2 + (y + 4) 2 = 16