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Lesson Menu Five-Minute Check (over Lesson 10–7) CCSS Then/Now New Vocabulary Key Concept: Equation of a Circle in Standard Form 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 Example 5: Intersections with Circles

Over Lesson 10–7 5-Minute Check 1 A.1 B.2 C.3 D.4 Find x.

Over Lesson 10–7 5-Minute Check 2 A.1 B.2 C.3 D.4 Find x.

Over Lesson 10–7 5-Minute Check 3 A.2 B.4 C.6 D.8 Find x.

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

Over Lesson 10–7 5-Minute Check 5 Find x in the figure. A.14 B. C. D.

CCSS Content Standards G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Mathematical Practices 2 Reason abstractly and quantitatively. 7 Look for and make use of structure.

Then/Now You wrote equations of lines using information about their graphs. Write the equation of a circle. Graph a circle on the coordinate plane.

Vocabulary compound locus

Concept

Example 1 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 = r 2 Equation of circle (x – 3) 2 + (y – (–3)) 2 =6 2 Substitution (x – 3) 2 + (y + 3) 2 = 36Simplify. Answer: (x – 3) 2 + (y + 3) 2 = 36

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

Example 1 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 A. Write the equation of the circle with a center at (2, –4) and a radius of 4.

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

Example 2 Write an Equation Using the Center and a Point 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 (x 1, y 1 ) = (–3, –2) and (x 2, y 2 ) = (1, –2) Simplify.

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

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

Example 3 Graph a Circle The equation of a circle is x 2 – 4x + y 2 + 6y = –9. State the coordinates of the center and the measure of the radius. Then graph the equation. Write the equation in standard form by completing the square. x 2 – 4x + y 2 + 6y= –9Original equation x 2 – 4x y 2 + 6y + 9 = – Complete the squares. (x – 2) 2 + (y + 3) 2 = 4Factor and simplify. (x – 2) 2 + [y – (–3)] 2 = 2 2 Write +3 as – (–3) and 4 as 2 2.

Example 3 Graph a Circle With the equation now in standard form, you can identify h, k, and r. (x – 2) 2 + [y – (–3)] 2 = 2 2 (x – h) 2 + [y – k] 2 = r 2 Answer:So, h = 2, y = –3, and r = 2. The center is at (2, –3), and the radius is 2.

Example 3 Which of the following is the graph of x 2 + y 2 –10y = 0? A.B. C.D.

Example 4 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, 1), and F(3, –4). Determine the location of a town equidistant from all three substations, and write an equation for the circle. UnderstandYou are given three points that lie on a circle. PlanGraph 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 Use Three Points to Write an Equation Solve Graph ΔDEF and construct the perpendicular bisectors of two sides.

Example 4 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 Use Three Points to Write an Equation CheckYou 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. 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.

Example 4 A.(3, 0) B.(0, 0) C.(2, –1) D.(1, 0) 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.

Example 5 Intersections with Circles Find the point(s) of intersection between x 2 + y 2 = 32 and y = x + 8. Graph these equations on the same coordinate plane.

Example 5 Intersections with Circles There appears to be only one point of intersection. You can estimate this point on the graph to be at about (–4, 4). Use substitution to find the coordinates of this point algebraically. x 2 + y 2 = 32Equation of circle. x 2 + (x + 8) 2 = 32Substitute x + 8 for y. x 2 + x x + 64 = 32Evaluate the square. 2x x + 32= 0 Simplify. x 2 + 8x + 16= 0Divide each side by 2. (x + 4) 2 = 0Factor. x= –4Take the square root of each side.

Example 5 Use y = x + 8 to find the corresponding y-value. (–4) + 8 = 4 The point of intersection is (–4, 4). Answer:(–4, 4) Intersections with Circles

Example 5 Find the points of intersection between x 2 + y 2 = 16 and y = –x. A.(2, –2) B.(2, 2) C.(–2, –2), (2, 2) D.(–2, 2), (2, –2)

End of the Lesson