Question

Which statement explains how you could use coordinate geometry to prove that quadrilateral ABCD is a rectangle? Shape ABCD is shown. Point A is at 5, 5. Point B is at 7, 3. Point C is at 3, 0. Point D is at 1, 2. 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 the slopes of consecutive sides are opposite reciprocals

Answer 1

To prove that a quadrilateral ABCD is a rectangle using coordinate geometry, show that opposite sides are congruent and parallel, and that adjacent sides are perpendicular by showing that their slopes are opposite reciprocals.

Step-by-step explanation:

To prove that quadrilateral ABCD is a rectangle using coordinate geometry, you need to demonstrate that both pairs of opposite sides are congruent (equal in length) and that consecutive sides are perpendicular to each other (implied by having slopes that are negative reciprocals of each other).

To show that segments AD and BC are congruent and parallel, you would calculate their lengths using the distance formula and show that the lengths are equal. Then, calculate the slope of each segment using the slope formula. If the slopes are equal, the segments are parallel.

Similarly, to prove segments AB and CD are congruent and parallel, follow the same process of computing the distance and slope for each segment. If the sides are congruent and the adjacent sides are found to be perpendicular (via the slope calculations), the quadrilateral is proven to be a rectangle.

Additionally, the slopes of adjacent sides should be opposite reciprocals, which would indicate that the angles between them are 90 degrees, meaning that the sides are perpendicular, which is a defining characteristic of a rectangle.