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.
Diagonal of a Rectangle
A rectangle has two diagonals. Each diagonal is a line segment connecting the opposite vertices of a rectangle.
Properties
Properties of Diagonals of a Rectangle
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:
Diagonal of a Rectangle Formula
Derivation
Derivation of Diagonal of Rectangle Formula
Applying Pythagorean theorem in ▭ ABCD, we get,
=> d2 = w2 + l2 , here d = diagonal, l = length, w = width
Taking square root on both sides,
d = √(w2 + l2)
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 = √(w2 + l2), here w = 8 cm, l = 12 cm = √(82 + 122) = √(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 = √(w2 + l2), here w = 19 inches, l = 44 inches = √(192 + 442) = 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?
Determine the length of the diagonal of a rectangle whose dimensions are 8 cm and 12 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.