Properties of a Rectangle

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.

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

The opposite sides of a rectangle are congruent.

A rectangle is also a parallelogram.

So, for a rectangle,
its opposite sides are congruent.

Properties of a parallelogram - Sides

Example 1

If ABCD is a rectangle, find AC. AB = 12, AD = 5.

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.

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

Find the value of x. The segment formed by bisecting diagonals: 5. Sides: x, 6.

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.