What is a Rectangle?
A rectangle is a type of quadrilateral or a four sided polygon. The opposite sides of a rectangle are congruent and parallel. Each angle of a rectangle is a right angle.
The figure in the image represents a rectangle ABCD.
So,
- AB = DC = w and AB∥DC
- AD = BC = l and AD∥BC
- ∠ABC = ∠BCD = ∠CDA = ∠DAB = 90°
Here, ‘l’ represents the length and ‘w’ represents the width of the rectangle respectively.
For the rectangle shown, AC and BD are its two diagonals. The diagonals of a rectangle are equal in length and they bisect each other. So,
- AC = BD and
- BO = OD = CO = OA
Diagonals of a Rectangle
As stated, a rectangle has two diagonals of equal length and each of these diagonals divides the rectangle into two right triangles. The length and width of the rectangle form the legs and the diagonals form the hypotenuse of each triangle.
Now, we have learned that a right triangle is associated with the Pythagorean theorem. Therefore, we can use the theorem to find the length of the diagonal of a rectangle.
Let us learn how to calculate the length of the diagonal of a rectangle.
Length of the Diagonal of a Rectangle
Consider the rectangle ABCD shown in the figure. The two diagonals form two right triangles each:
- Triangles ABC and ADC formed by diagonal AC
- Triangles ABD and BCD formed by diagonal BD
In right triangle ABC, right angled at B:
- Diagonal AC, is the hypotenuse
- Sides AB and BC, are the legs
So,
\(AB^{2} + BC^{2} = AC^{2}\) [Applying Pythagorean theorem]
\(l^{2} + w^{2} = d^{2}\) [Substitute l for AB, w for BC and d for AC]
\(d = \sqrt{l^{2} + w^{2}}\) [Solve for d]
Since, both the diagonals of a rectangle are equal in length, so,
Diagonal BD, \(d = \sqrt{l^{2} + w^{2}}\)
So, here we have the formula to calculate the length of the diagonal of a rectangle:
Length of diagonal of a rectangle, \(d = \sqrt{l^{2} + w^{2}}\)
Hence this formula can be used to determine the length of the diagonal as well as the lengths of the sides of a rectangle.
Solved Examples
Example 1: Find the length of each diagonal of a rectangle with dimensions 5 units and 12 units.
Solution:
\(d = \sqrt{l^{2} + w^{2}}\) [Formula for the length of diagonal of a rectangle]
\(d = \sqrt{l2^{2} + 5^{2}}\) [Substitute values]
\(d = \sqrt{44 + 25}\) [Square of numbers]
\(d = \sqrt{169}\) [Add]
d = 13 units [Positive square root of 169]
So, the length of each diagonal of the rectangle is 13 units.
Example 2: Find the measure of length of the rectangle shown in the figure.
Solution:
From the figure:
d = 5 cm
w = 3 cm
\(d = \sqrt{l^{2} + w^{2}}\) [Formula for the length of diagonal of a rectangle]
\(5 = \sqrt{l^{2} + 3^{2}}\) [Substitute values]
\(5 = \sqrt{l^{2} + 9}\) [Square of number]
\(25 = l^{2} + 9\) [Square on both sides]
\(l^{2} = 25 – 9\) [Solve for \(l^{2} \)]
\(l^{2} = 16\) [Simplify]
\(l = 4\text{ }cm\) [Positive square root of 16]
So, the measure of the length of the rectangle is 4 centimeters.
Example 3: The size of a tablet screen is the length of its diagonal. Find the size of a tablet screen with dimensions 16 centimeters by 14 centimeters.
Solution:
\(d = \sqrt{l^{2} + w^{2}}\) [Formula for the length of diagonal of a rectangle]
\(d = \sqrt{16^{2} + 14^{2}}\) [Substitute values]
\(d = \sqrt{256 + 196}\) [Square of numbers]
\(d = \sqrt{452}\) [Add]
d = 21.26 cms [Positive square root of 400]
So, the size of the tablet screen is 21.26 centimeters.
A diagonal is a straight line connecting the non-adjacent vertices of a polygon.
The three sides of the two triangles formed by the diagonals of a rectangle are length, width and the diagonal of the rectangle. So, the two triangles have corresponding sides of equal length. Therefore, the two triangles are congruent.
The angles formed at the intersection of the diagonals of a rectangle are a pair of acute angles, and a pair of obtuse angles.