Geometry (A): READ: Rectangles and Squares Reading

Rhombi, Rectangles, and Squares

Learning Objectives

Identify the relationship between the diagonals in a rectangle.
List the properties of a rectangle.
List the properties of a square.

Introduction

Now that you have a much better understanding of parallelograms, you can begin to look more carefully into certain types of parallelograms. This lesson explores two very important types of parallelograms—rectangles and squares. Remember that all of the rules that apply to parallelograms still apply to rectangles and squares. In this lesson, you’ll learn about rules specific to these shapes that are not true for all parallelograms.

Diagonals in a Rectangle

Recall from previous lessons that the diagonals in a parallelogram bisect each other. You can prove this with congruence of triangles within the parallelogram. In a rectangle, there is an even more special relationship between the diagonals. The two diagonals in a rectangle will always be congruent. We can show this using the distance formula on a coordinate grid.

Example 1

Use the distance formula to demonstrate that the two diagonals in the rectangle below are congruent.

To solve this problem, you need to find the lengths of both diagonals in the rectangle. First, draw line segments that connect the vertices of the rectangle. So, draw a segment from $(-2, 3)$ to $(4, 6)$ and from $(-2, 6)$ to $(4, 3)$ .

You can use the distance formula to find the length of the diagonals. Diagonal $\overline {BD}$ goes from $B(-2, 3)$ to $D(4, 6)$ .

$BD &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\ &=\sqrt{(-2 - 4)^2 + (3 - 6)^2}\\ &=\sqrt{(-6)^2 + (-3)^2}\\ &=\sqrt{36 + 9}\\ &=\sqrt{45}$

Next, find the length of diagonal $\overline {AC}$ . That diagonal goes from $A(-2, 6)$ to $C(4, 3)$ .

$AC &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\ &=\sqrt{(4 - (-2))^2 + (3 - 6)^2}\\ &=\sqrt{(6)^2 + (-3)^2}\\ &=\sqrt{36 + 9}\\ &=\sqrt{45}$

So, $BD = AC = \sqrt{45}$ . In this example, the diagonals are congruent. Are the diagonals of rectangles always congruent? The answer is yes.

Theorem: The diagonals of a rectangle are congruent

The proof of this theorem relies on the definition of a rectangle (a quadrilateral in which all angles are congruent) as well as the property that rectangles are parallelograms.

Given: Rectangle $RECT$
Prove: $\overline {RC} \cong \overline {ET}$

Statement	Reason
1. $RECT$ is a rectangle	1. Given
2. $\angle {RTC} \cong \angle {TCE}$	2. Definition of a rectangle
3. $\overline {RT} \cong \overline {EC}$	3. Opposite sides of a parallelogram are $\cong$
4. $\overline {TC} \cong \overline {TC}$	4. Reflexive Property of $\cong$
5. $\triangle {RTC} \cong \triangle {ECT}$	5. SAS Congruence Postulate
6. $\overline {RC} \cong \overline {ET}$	6. Definition of congruent triangles (corresponding parts of congruent triangles are congruent)

List of Properties for a Rectangle

1) Opposite sides are congruent.
2) Opposite angles are congruent.
3) Consecutive angles are supplementary.
4) Diagonals bisect each other.
5) Diagonals separate onto two congruent triangles.
6) All four angles are congruent. (90 degrees)
7) Diagonals are congruent.

~The first five properties the properties of a parallelogram. Since a rectangle is a parallelogram, it has all the properties of a parallelogram. Properties 6 and 7 and specific to rectangles.

Squares

Squares are a combination of a rectangle and a rhombus. It is a parallelogram where all four angles are congruent like a rectangle, but all four sides are also congruent like a rhombus. Therefor, a square follows all the properties of a rectangle and a rhombus.

Properties of a Square
1) Opposite sides are congruent.
2) Opposite angles are congruent.
3) Consecutive angles are supplementary.
4) Diagonals bisect each other.
5) Diagonals separate onto two congruent triangles.
6) All four angles are congruent. (90 degrees)
7) Diagonals are congruent
8) All four sides are congruent.
9) Diagonals are perpendicular.
10) Diagonals are angle bisectors.

Lesson Summary

In this lesson, we explored rhombi, rectangles, and squares. Specifically, we have learned:

How to identify and prove the relationship between the diagonals in a rectangle.
List all the properties of a rectangle.
List all the properties of a square.

It is helpful to be able to identify specific properties in quadrilaterals. You will be able to use this information in many different ways.

The following questions are for your own review. The answers are listed below to help you check your work and understanding.