Exercise 1 Page 416

a Since the angles of a rectangle are right angles, we can place

P Q R S

in the coordinate plane with two sides on the coordinate axes.

b Let's think about the relationship between the coordinates of

P,

Q,

R,

and

S .

Vertex $Q$ is on the $y -$ axis, so the $x -$ coordinate is $0 .$
Vertex $S$ is on the $x -$ axis, so the $y -$ coordinate is $0 .$
Side $\overline{R S}$ is parallel to the $y -$ axis, so the $x -$ coordinate of $R$ and $S$ are the same.
Side $\overline{Q R}$ is parallel to the $x -$ axis, so the $y -$ coordinate of $Q$ and $R$ are the same.

Let's use variables to name the coordinates of $Q,$ $R,$ and $S .$

These are just example variables, so your variable choices may differ.

c It is given that

P Q R S

is a rectangle, and the claim is that the diagonals are congruent.

Given : Prove : P Q R S is a rectangle \overline{P R} ≅ \overline{Q S}

d To show that two segments are congruent, it is enough to show that they have the same lengths. In a coordinate proof we can use the Distance Formula to express the length of a segment in terms of the coordinates of the endpoints.