Question

If the diagonal of a rectangle is 17 cm and its perimeter is 46 cm, the area of the rectangle is

(a) 100 cm² (b) 110 cm² (c) 120 cm² (d) 240 cm²

Answer 1

Let l and b be the length and breadth of the rectangle, where diagonal = 17 cm and perimeter = 46 cm.
So, as per the question
Perimeter = 2 (l + b)
$\Rightarrow$ 46 = 2 (l + b)
$\Rightarrow$ l + b = 23 ..... (i)
Now, in the triangle formed by the adjacent sides and one diagonal of the rectangle, using Pythagoras theorem, we have
l² + b² = (diagonal)²
$\Rightarrow$ l² + b² = 17²
$\Rightarrow$ l² + (23 − l)² = 17² [From (i)]
$\Rightarrow$ l² +l² + 23² − 46l = 289
$\Rightarrow$ 2l² + 529 − 46l = 289
$\Rightarrow$ 2l² − 46l + 240 = 0
$\Rightarrow$ l² − 23l + 120 = 0
$\Rightarrow$ l² − 15l − 8l + 120 = 0
$\Rightarrow$ l(l − 15) − 8(l − 15) = 0
$\Rightarrow$ (l − 15) (l − 8) = 0
$\Rightarrow$ l = 15 cm or l = 8 cm
If l = 15 cm, then from (i), b = 23 − 15 = 8 cm.
If l = 8 cm, then from (i), b = 23 − 8 = 23 cm.
Therefore
Area of the rectangle = l $\text{×}$ b = 15 $\text{×}$ 8 = 120 cm2
Hence, the correct option is (c).