Question

The perimeter of a triangle is 50 cm. One side of the triangle is 4 cm longer than the smallest side and the third side is 6 cm less than twice the smallest side. Find the area of the triangle.

• A. 170 cm${}^2$
• B. $20\sqrt{30}{\text{cm}}^2$
• C. $17\sqrt{30}{\text{cm}}^2$
• D. 20 cm${}^2$

Answer 1

The correct option is B $20\sqrt{30}{\text{cm}}^2$

Given, Perimeter of the triangle is 50 cm.

Let, smaller side of the triangle be 'a' cm.
So, the other sides are 'a + 4' and '2a - 6'

$\because$ The perimeter of a triangle is sum of all its sides

$\therefore$ a + (a + 4) + (2a - 6) =50

$\Rightarrow$ 4a - 2 = 50

$\Rightarrow$ a = 13

$\therefore$ Sides of the triangle are 13 cm, 17 cm, 20 cm

$\text{Semiperimeter (s)}=\frac{20+17+13}{2}=\frac{50}{2}=25$

Using herons formula:
$\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}=\left(\sqrt{25(25-13)(25-17)(25-20)}\right)$
$=20\sqrt{30}{\text{cm}}^2$