The perimeter of a triangle is 300 m. If its sides are in the ratio 3

1.	The perimeter of a triangle is 300 m. If its sides are in the ratio 3 : 5 : 7. Find the area of the triangle.
Answer» Sides of triangle are in ratio: 3 : 5 : 7 a = 3x, b = 5x, c = 7x Since the perimeter of a triangle is given by: a+b+c = perimeter 3x +5x+7x = 300 x = \(\frac{300}{15}\) = 20 x = 20 Therefore sides of the triangle are: a = 3x = 3 x 20 = 60, b = 5x = 5 x 20 = 100, c = 7x = 7 x 20 = 140 When a, b and c are the sides of triangle and s is the semi-perimeter, then its area is given by: A = \(\sqrt{s(s-a)(s-b)(s-c)}\) where s = \(\frac{a+b+c}{2}\)[Heron’s Formula] s = \(\frac{a+b+c}{2}\) = \(\frac{60+100+140}{2}\) = 150 A = \(\sqrt{150(150-60)(150-100)(150-140)}\) A = \(\sqrt{150\times 90\times50 \times10}\) = \(\sqrt{15\times 9\times5 \times10000}\) = \(\sqrt{15\times 3\times3 \times5 \times10000}\) = \(\sqrt{15\times 3\times15 \times10000}\) = \(15\times 100\sqrt{3}\) = \(1500\sqrt{3}\) m²

The perimeter of a triangle is 300 m. If its sides are in the ratio 3 : 5 : 7. Find the area of the triangle.

Answer»

Sides of triangle are in ratio: 3 : 5 : 7

a = 3x, b = 5x, c = 7x

Since the perimeter of a triangle is given by:

a+b+c = perimeter

3x +5x+7x = 300

x = \(\frac{300}{15}\) = 20

x = 20

Therefore sides of the triangle are:

a = 3x = 3 x 20 = 60,

b = 5x = 5 x 20 = 100,

c = 7x = 7 x 20 = 140

When a, b and c are the sides of triangle and s is the semi-perimeter, then its area is given by:

A = \(\sqrt{s(s-a)(s-b)(s-c)}\) where s = \(\frac{a+b+c}{2}\)[Heron’s Formula]

s = \(\frac{a+b+c}{2}\) = \(\frac{60+100+140}{2}\) = 150

A = \(\sqrt{150(150-60)(150-100)(150-140)}\)

A = \(\sqrt{150\times 90\times50 \times10}\) = \(\sqrt{15\times 9\times5 \times10000}\)

= \(\sqrt{15\times 3\times3 \times5 \times10000}\) = \(\sqrt{15\times 3\times15 \times10000}\)

= \(15\times 100\sqrt{3}\) = \(1500\sqrt{3}\) m²