Question

Which statement explains how you could use coordinate geometry to prove that quadrilateral ABCD is a rectangle?

a coordinate plane with quadrilateral ABCD at A negative 2 comma 0, B 0 comma negative 2, C negative 3 comma negative 5, D negative 5 comma negative 3

Prove that segments AD and AB are congruent and parallel.
Prove that opposite sides are congruent and that the slopes of consecutive sides are equal.
Prove that segments BC and CD are congruent and parallel.
Prove that opposite sides are congruent and that the slopes of consecutive sides are opposite reciprocals.

Answer 1

Option 4: Prove that opposite sides are congruent and that the slopes of consecutive sides are opposite reciprocals

Explanation:

Let's solve this step-by-step:

A rectangle is a shape with four 90 degree angles, opposite sides are equal in length, the diagonals bisect each other of equal length, and the slopes of the consecutive sides are opposite reciprocals

The first option cannot work because they are not congruent or parallel. They have different slopes and different lengths.

Option 1:

1. To find if AD and AB are congruent, you need to find if they have equivalent length. To do this, we can see that AD and AB have right angles. Due to this, we can use the pythagorean theorem.

We find both the legs for AD from (-2,0) to (-5,-3) and use the pythagorean theorem. Find the vertical and horizontal distance from -2 to -5 and 0 to -3.

Vertical: 3

Horizontal: 3

3^2 + 3^2 = sqrt 18

Now, we find the legs for AB from (-2,0) to (0,-2).

Vertical: 2

Horizontal: 2

2^2 + 2^2 = sqrt 8

This does not work because they are not congruent so the first option doesn't work.

Option 2:

The opposite sides of this rectangle are AB and CD. We can use the pythagorean theorem to figure out that both have a vertical and horizontal distance of 2.

Apply the pythagorean theorem: 2^2 + 2^2 = sqrt 8.

Both have congruent sides. The problem is that the slopes of the consecutive sides are not equivalent so they are not congruent. This does not work.

Option 3:

We have already gone through so much work to know that BC and CD are not parallel because they do not have the same slope. They also are not congruent because when you apply the pythagorean theorem, BC = sqrt 18 and CD = square root 8.

This does not work.

Option 4:

We have already proven that the opposite sides are congruent so we can move on to the second case. Now, let's use CD and BC to compare each slope.

The formula for slope is y^2 - y^1 / x^2 - x^1.

CD : (-3,-5) (-5,-3)

-3 + 5/ -5 + 3 = -1.

BC: (0,-2)(-3,-5)

-5 + 2/ -3 - 0 = 1

1 is the opposite reciprocal of -1 so Option 4 or Option D