The perimeter of 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.
The perimeter of 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.
Let ABC be any triangle with perimeter 50 cm.
Let the smallest side of the triangle be x.
Then the other sides be x + 4 and 2x − 6.
Now,
x + x + 4 + 2x − 6 = 50 (∵ perimeter is 50 cm)
⇒ 4x − 2 = 50
⇒ 4x = 50 + 2
⇒ 4x = 52
⇒ x = 13
∴ The sides of the triangle are of length 13 cm, 17 cm and 20 cm.
∴ Semi-perimeter of the triangle is
$s=\frac{13+17+20}{2}=\frac{50}{2}=25 \mathrm{~cm}$
∴ By Heron's formula,
Area of $\Delta A B C=\sqrt{s(s-a)(s-b)(s-c)}$
$=\sqrt{25(25-13)(25-17)(25-20)}$
$=\sqrt{25(12)(8)(5)}$
$=20 \sqrt{30} \mathrm{~cm}^{2}$
Hence, the area of the triangle is $20 \sqrt{30} \mathrm{~cm}^{2}$.