Circles Unit 6 Lesson 7 Circles in the

Circles Unit 6: Lesson 7 Circles in the Coordinate Plane Holt Geometry Texas © 2007

Objectives and Student Expectations � TEKS: G 2 B, G 4 A, G 5 A � The student will make conjectures about angles, lines, polygons and determine the validity of the conjectures. � The student will select an appropriate representation in order to solve problems. � The student will use geometric patterns to develop algebraic expressions.

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.

Example: 1 Write the equation of each circle. J with center J (2, 5) and radius 4 (x – h)2 + (y – k)2 = r 2 (x – 2)2 + (y – 5)2 = 42 (x – 2)2 + (y – 5)2 = 16 Equation of a circle Substitute 2 for h, 2 for k, and 4 for r. Simplify.

Example: 2 Write the equation of each circle. K that passes through J(6, 4) and has center K(1, – 8) Distance formula. Simplify. Substitute 1 for h, – 8 for (x – + (y – = k, and 13 for r. (x – 1)2 + (y + 8)2 = 169 Simplify. 1)2 (– 8))2 132

Example: 3 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 = 82 x 2 + (y + 3)2 = 64 Equation of a circle Substitute 0 for h, – 3 for k, and 8 for r. Simplify.

Example: 4 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 = 42 (x – 2)2 + (y + 1)2 = 16 Substitute 2 for h, – 1 for k, and 4 for r. Simplify.

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.

Example: 5 Graph x 2 + y 2 = 25 Step 1 Find the center of the circle. Center = (0, 0) Step 2 Find the radius of the circle. Since the radius is 5, use ± 5. Step 3 Plot the points and connect them to form a circle.

Example: 6 Graph (x-5)2 + (y-3)2 = 16 Step 1 Find the center of the circle. Center = (5, 3) Step 2 Find the radius of the circle. Since the radius is 4, use ± 4. Step 3 Plot the points and connect them to form a circle.

Example: 7 Graph (x+6)2 + (y+4)2 = 9 Step 1 Find the center of the circle. Center = (-6, -4) Step 2 Find the radius of the circle. Since the radius is 3, use ± 3. Step 3 Plot the points and connect them to form a circle.

Example: 8 Graph (x+7)2 + (y-2)2 = 1 Step 1 Find the center of the circle. Center = (-7, 2) Step 2 Find the radius of the circle. Since the radius is 1, use ± 1. Step 3 Plot the points and connect them to form a circle.

Example: 9 Write the equation and graph the circle. Center (6, -5) and r=2 Step 1 Graph the circle. Step 2 Develop the equation for the circle. (x-6)2 + (y+5) 2 = 4

Example: 10 Write the equation and graph the circle. Center (0, 7) and r=3 Step 1 Graph the circle. Step 2 Develop the equation for the circle. x 2 + (y-7) 2 = 9

Given 3 points that are on the circumference of the circle, find the center of the circle and the equation of the circle. Recall from previous work on triangles that the circumcenter ________________ is the point that is equidistant from the vertices of the triangle. Graph the points, draw the triangle, and draw the perpendicular bisectors to find the center of the circle.

Example: 11 Note: this graph paper is not “square”. Find the center and write the equation of the circle given points A(-11, 11), B(-5, -7), and C(3, -3).

Plot points and draw the triangle. Find the midpoint of each side.

Draw the perpendicular bisectors for each side using slopes. The concurrent point is the circumcenter at (-5, 3), the center of the circle.

The radius is 10 from center down to point at (-5, -7). Plot more points: up, left, and right 10 from center and draw the circle. Remember the graph paper is not “square”.