# Properties of a Rectangle

How to solve the rectangle problems by using its properties: definition, properties of its sides and diagonals, examples, and their solutions.

## Definition

A rectangle is a parallelogram

whose interior angles are all right angles: 90º.

So a rectangle

has all the properties of a parallelogram.

Properties of a parallelogram

## Property: Sides

A rectangle is also a parallelogram.

So, for a rectangle,

its opposite sides are congruent.

Properties of a parallelogram - Sides

## Example 1

The given figure is a rectangle.

So its two opposite sides are congruent.*AB* and *CD* are the opposite sides.

And *AB* = 12.

So *CD* = 12.

See △*ADC*.

It's a right triangle.

And starting from the shortest side,

the sides are (5, 12, *AC*).

So △*ADC* is a (5, 12, 13) right triangle.

Pythagorean triples

So *AC* = 13.

## Property: Diagonals

For a rectangle,

the segments formed by the diagonals

are all congruent.

This is true because

the diagonals of a rectangle are congruent

and the diagonals of a parallelogram

bisect each other.

Properties of a parallelogram - Diagonals

## Example 2

The given figure is a rectangle.

So the segments formed by the diagonals

are all congruent.

Write the lengths of the inclined segments: 5.

See this right triangle.

It's a right triangle.

And starting from the shortest side,

the sides are (6, *x*, 10).

So The given triangle is similar to

the (3, 4, 5) right triangle.

Pythagorean triples

Since these two triangles are similar,

their sides are proportional.

So *x*/4 = 10/2.

Similar triangles

10/2 = 5

Multiply 4 on both sides.

Then *x* = 8.