Question

The vertices of a quadrilateral are A(-1,6),B(-2,4),C(2,2), and D(3,4). Write a paragraph proof to determine whether quadrilateral ABCD is a rectangle. 15px

Answer 1

Determining the slopes of each side, we get the slope of $\overline{AB}$ is $2$, the slope of $\overline{BC}$ is -1/2, the slope of $\overline{CD}$ is 2, and the slope of $\overline{AD}$ is -1/2. Since the slopes of sides AB and CD and BC and AD are equal, it follows that $\overline{AB} \parallel \overline{CD}$ and $\overline{BC} \parallel \overline{AD}$. Thus, ABCD is a parallelogram because it is a quadrilateral with two pairs of opposite congruent sides. However, we can also note that the slope of side AB is the negative reciprocal of that of sides BC, and thus $\overline{AB} \perp \overline{BC}$. Using the fact that perpendicular lines form right angles, we can conclude that $\angle ABC$ is a right angle, and since ABCD is thus a parallelogram with a right angle, it must also be a rectangle.