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 y-axis. Label the vertices A, B, C, and D.

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:

The x-coordinate of C is also a. The y-coordinate is b because the side is b units long.

Position and label a parallelogram with sides a and b units long on the coordinate plane.

Sample answer:

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 and the y-coordinate of D is

Answer:

Name the missing coordinates for the rhombus.

Answer:

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.

Proof:

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

Find the slopes of

slope of

slope of

slope of

slope of

A segment with slope 0 is perpendicular to a segment with undefined slope. Therefore, consecutive sides of this quadrilateral are perpendicular. Since consecutive sides 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. Since all four sides are congruent and opposite

sides are parallel, MNPQ is a rhombus.

Proof:

Write a coordinate proof to prove that the supports of a platform lift are parallel.

Prove:

Proof:

Given: A(5, 0), B(10, 5), C(5, 10), D(0, 5)

Since have the same slope, they are parallel.

Write a coordinate proof to prove that the crossbars of a child safety gate are parallel.

Prove:

Proof: Since have the same slope, they are parallel.

Given: A(–3, 4), B(1, –4), C(–1, 4), D(3, –4)