1. In the figure, ABCD is an isosceles trapezoid with median EF. Find mD if mA = 110.

2. Find x if AD = 3x2 – 5 and BC = x2 + 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.

• Position and label quadrilaterals for use in coordinate proofs.

• Prove theorems using coordinate proofs.

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.

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.

A. A

B. B

C. C

D. 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.

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.

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.

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.

Coordinate Proof

Proof:

By the Midpoint Formula, the coordinates of M, N, P, and Q are as follows.

Find the slopes of

Coordinate Proof

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.

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.

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:

1. A

2. B

3. C

4. D

A.

B.

C.

D.

Which expression would be the lengths of the four sides of MNPQ?

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)

A. A

B. B

C. C

D. D

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)