1. Lesson 7 Menu 1.In the figure, ABCD is an isosceles trapezoid with median EF. Find m  D if m  A = 110. 2.Find x if AD = 3x 2 – 5 and BC = x 2 + 27. 3.Find y if AC = 9(2y – 4) and BD = 10y + 12. 4.Find EF if AB = 10 and CD = 32. 5.Find AB if AB = r + 18, CD = 6r + 9, and EF = 4r + 10.

2. Lesson 7 MI/Vocab Position and label quadrilaterals for use in coordinate proofs. Prove theorems using coordinate proofs.

3. Lesson 7 Ex1 Positioning a Square POSITIONING A RECTANGLE Position and label a rectangle with sides a and b units long on the coordinate plane. The y-coordinate of B is 0 because the vertex is on the x-axis. Since the side length is a, the x-coordinate is a. Let A, B, C, and D be vertices of a rectangle with sides a units long, and sides b units long. Place the square with vertex A at the origin, along the positive x-axis, and along the positive y-axis. Label the vertices A, B, C, and D.

4. Lesson 7 Ex1 Positioning a Square Sample answer: D is on the y-axis so the x-coordinate is 0. Since the side length is b, the y-coordinate is b. The x-coordinate of C is also a. The y-coordinate is 0 + b or b because the side is b units long.

5. A.A B.B C.C D.D Lesson 7 CYP1 Position and label a square with sides a units long on the coordinate plane. Which diagram would best achieve this? A.B. C.D.

6. Lesson 7 Ex2 Find Missing Coordinates Name the missing coordinates for the isosceles trapezoid. Answer: D(b, c) The legs of an isosceles trapezoid are congruent and have opposite slopes. Point C is c units up and b units to the left of B. So, point D is c units up and b units to the right of A. Therefore, the x-coordinate of D is 0 + b, or b, and the y-coordinate of D is 0 + c, or c.

7. Lesson 7 CYP2 1.A 2.B 3.C 4.D A.C(c, c) B.C(a, c) C.C(a + b, c) D.C(b, c) Name the missing coordinates for the parallelogram.

8. Lesson 7 Ex3 Coordinate Proof Place a rhombus on the coordinate plane. Label the midpoints of the sides M, N, P, and Q. Write a coordinate proof to prove that MNPQ is a rectangle. The first step is to position a rhombus on the coordinate plane so that the origin is the midpoint of the diagonals and the diagonals are on the axes, as shown. Label the vertices to make computations as simple as possible. Given: ABCD is a rhombus as labeled. M, N, P, Q are midpoints. Prove: MNPQ is a rectangle.

9. Lesson 7 Ex3 Coordinate Proof Proof: By the Midpoint Formula, the coordinates of M, N, P, and Q are as follows. Find the slopes of

10. Lesson 7 Ex3 Coordinate Proof

11. Lesson 7 Ex3 Coordinate Proof A segment with slope 0 is perpendicular to a segment with undefined slope. Therefore, consecutive sides of this quadrilateral are perpendicular. MNPQ is, by definition, a rectangle.

12. Lesson 7 CYP3 Place an isosceles trapezoid on the coordinate plane. Label the midpoints of the sides M, N, P, and Q. Write a coordinate proof to prove that MNPQ is a rhombus. Given: ABCD is an isosceles trapezoid. M, N, P, and Q are midpoints. Prove: MNPQ is a rhombus.

13. Lesson 7 CYP3 The coordinates of M are (–3a, b); the coordinates of N are (0, 0); the coordinates of P are (3a, b); the coordinates of Q are (0, 2b). Since opposite sides have equal slopes, opposite sides are parallel and MNPQ is a parallelogram. The slope of The slope of is undefined. So, the diagonals are perpendicular. Thus, MNPQ is a rhombus. Proof:

14. 1.A 2.B 3.C 4.D Lesson 7 CYP3 A. B. C. D. Which expression would be the lengths of the four sides of MNPQ?

15. Lesson 7 Ex4 Properties of Quadrilaterals Proof: Since have the same slope, they are parallel. Write a coordinate proof to prove that the supports of a platform lift are parallel. Prove: Given: A(5, 0), B(10, 5), C(5, 10), D(0, 5)

16. A.A B.B C.C D.D Lesson 7 CYP4 A.slopes = 2 B.slopes = –4 C.slopes = 4 D.slopes = –2 Prove: Given: A(–3, 4), B(1, –4), C(–1, 4), D(3, –4)