Here we will discuss about the perimeter and area of a rectangle and some of its geometrical properties.
Perimeter of a rectangle (P) = 2(length + breadth) = 2(l + b)
Area of a rectangle (A) = length × breadth = l × b
Diagonal of a rectangle (d) = \(\sqrt{(\textrm{length})^{2}+(\textrm{breadth})^{2}}\)
= \(\sqrt{\textrm{l}^{2}+\textrm{b}^{2}}\)
Length of a rectangle (l) = \(\frac{\textrm{area}}{\textrm{breadth}} = \frac{A}{b}\)
Breadth of a rectangle (b) = \(\frac{\textrm{area}}{\textrm{length}} = \frac{A}{l}\)
Some geometrical properties of a rectangle:
In the rectangle PQRS,
PQ = SR, PS = QR, QS = PR;
OP = OR = OQ = OD;
∠PSC = ∠QRS = ∠RQP = ∠qps = 90°.
Also, PR^{2} = PS^{2} + SR^{2}; [by Pythagoras’ theorem)
and QS2 = QR2 + SR2; [by Pythagoras’ theorem)
Area of the ∆PQR = Area of the ∆PSQ = Area of the ∆QRS = Are of the ∆PSR
= \(\frac{1}{2}\) (Area of the rectangle PQRS).
Solved Examples on Perimeter and Area of a Rectangle:
1. The area of a rectangle whose sides are in the ratio 4:3 is 96 cm\(^{2}\). What is the perimeter of the square whose each side is equal in length to the diagonal of the rectangle?
Solution:
As the sides og the rectangle are in the ratio 4:3, let the sides be 4x and 3x respectively.
Then, the area of the rectangle = 4x ∙ 3x = 96 cm\(^{2}\)
Therefore, 12x\(^{2}\) = 96 cm\(^{2}\)
or, x\(^{2}\) = 8 cm\(^{2}\)
Therefore, x = 2√2 cm
Now, the length of a diagonal of the square = \(\sqrt{(4x)^{2} + (3x)^{2}}\)
= \(\sqrt{25x^{2}}\)
= 5x
Therefore, the perimeter of the square = 4 × side
= 4 × 5x
= 20x
= 20 × 2√2 cm
= 40√2 cm
= 40 × 1.41 cm
= 56.4 cm
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