The length of the sides of a triangle are in the ratio 3:4::5 and its perimeter is 144 cm - Maths - Heron\s Formula

Question

The length of the sides of a triangle are in the ratio 3:4::5
and its perimeter is 144 cm Find the area of the triagnle and the
height corresponding to the longest side

Ankush Jain · Accepted Answer

Given, the length of sides of the triangle is the ratio 3 : 4 : 5.

and perimeter = 144cm

⇒ 3x + 4x + 5x =144

⇒ 12x =144

⇒ x =12cm

∴ shortest side = 3x = 3 ×12 = 36cm

side of medium length = 4x = 4 × 12 = 48cm

longest side = 5x = 5 ×12 = 60cm

Again s = semi-perimeter of Δ = $\frac{144}{2}$ = 72cm

Using, heron's formula, area of Δ

$= \sqrt{s (s - a) (s - b) (s - c)} = \sqrt{72 (72 - 36) (72 - 48) (72 - 60)} = \sqrt{72 \times 36 \times 24 \times 12} = \sqrt{36 \times 2 \times 36 \times 12 \times 2 \times 12} = 36 \times 12 \times 2 {cm}^{2} = 36 \times 24 {cm}^{2} = 864 {cm}^{2} ...... (1) Also, area of Δ = \frac{1}{2} \times base \times height = \frac{1}{2} \times 60 \times h [Base = 60 cm; Given, the height should corresponds to the longest side] = 30 h ...... (2)$

Equating (1) and (2), we get

$30 h = 864 \Rightarrow h = 28.8 cm$

Hence, the height of triangle corresponding to longest side is 28.8 cm.

30

The length of the sides of a triangle are in the ratio 3:4::5 and its perimeter is 144 cm. Find the area of the triagnle and the height corresponding to the longest side.