While learning how to find out the area and perimeter of a rectangle, we came across the term diagonal many a times. In this article, we will learn about the diagonal and how to calculate it.

What is the Diagonal of a Rectangle

The diagonal of a rectangle is the line segments linking opposite vertices or corners of the rectangle.

A rectangle has two diagonals. Each diagonal is a line segment connecting the opposite vertices of a rectangle.

Properties

The 2 diagonals are congruent (equal in length); here AC and BD are the diagonals & AC = BD

The diagonals bisect each other equally; here AC & BD bisect each other

Each diagonal divides a rectangle into 2 congruent right-angled triangles; here AC divides ▭ ABCD into 2 ∆ABC and ∆ADC; BD divides ▭ ABCD into ∆BCD and ∆BAD

The diagonals are not perpendicular to each other; here diagonals AC & BD are not perpendicular

The adjacent central angles at the point of intersection are not equal, but the opposite central angles are equal; here ∠AOB = ∠COD & ∠BOC = ∠AOD, but ∠AOB ≠ ∠AOD or ∠BOC, & ∠BOC ≠ ∠COD or ∠AOB

Formula

The formula to find the diagonal of a rectangle is given below:

Derivation

Applying Pythagorean theorem in ▭ ABCD, we get,

=> d^{2} = w^{2} + l^{2} , here d = diagonal, l = length, w = width

Taking square root on both sides,

d = √(w^{2} + l^{2})

Thus we get the formula to calculate the diagonal of a rectangle.

Let us see how to calculate the diagonals of rectangles by solving some examples.

Solved Examples

Determine the length of the diagonal of a rectangle whose dimensions are 8 cm and 12 cm.

Solution:

As we know, d = √(w^{2} + l^{2}), here w = 8 cm, l = 12 cm = √(8^{2} + 12^{2}) = √(64 + 144) = √208 ≈ 14.42 cm

If the size of a television screen is its diagonal, then find the size of a television whose dimensions are 19 inches and 44 inches.

Solution:

As we know, d = √(w^{2} + l^{2}), here w = 19 inches, l = 44 inches = √(19^{2} + 44^{2}) = 47.93 ≈ 48 inches In the practical world, television sizes are stated as the closest approximate value.

FAQs

Q1. Do the diagonals of a rectangle bisect each other?

Ans: Yes.

Q2. Do the diagonals of a rectangle bisect the angles?