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Lesson 7 Menu

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

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Lesson 7 Menu

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  1. In the figure, ABCD is an isosceles trapezoid with median EF. Find mD if mA = 110. • Find x if AD = 3x2 – 5 and BC = x2 + 27. • 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. Lesson 7 Menu

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

  3. 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. 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. Lesson 7 Ex1

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

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

  6. Find Missing Coordinates Name the missing coordinates for the isosceles trapezoid. 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. Answer:D(b, c) Lesson 7 Ex2

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

  8. 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. Lesson 7 Ex3

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

  10. Coordinate Proof Lesson 7 Ex3

  11. 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. Lesson 7 Ex3

  12. 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. Lesson 7 CYP3

  13. Proof: 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. Lesson 7 CYP3

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

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

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

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