The lengths of the sides of a triangle are in the ratio 3 : 4 : 5 and

1.	The lengths of the sides of a triangle are in the ratio 3 : 4 : 5 and its perimeter is 144 cm. Find the area of the triangle and the height corresponding to the longest side
Answer» Sides of triangle are in ratio: 3 : 4 : 5 a = 3 x, b = 4x, c = 5x Since the perimeter of a triangle is given by: a+b+c = perimeter. 3x+4x+5x = 144 x = \(\frac{144}{12}\) = 12 x = 12 Therefore sides of the triangle are: a = 3x = 3 x 12 = 36, b = 4x =4 x 12= 48, c = 5x = 5 x 12= 60 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 = \(\frac{a+b+c}2\) = \(\frac{36+48+60}2\) = 72 A = \(\sqrt{72(72-36)(72-48)(72-60)}\) A = \(\sqrt{72\times 36 \times24 \times12}\) = 864 cm² Area of triangle = \(\frac{1}2\)(Base x Altitude) 864 = \(\frac{1}2\)(60 x Altitude) Altitude = 28.8 cm

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

Answer»

Sides of triangle are in ratio: 3 : 4 : 5

a = 3 x, b = 4x, c = 5x

Since the perimeter of a triangle is given by:

a+b+c = perimeter.

3x+4x+5x = 144

x = \(\frac{144}{12}\) = 12

x = 12

Therefore sides of the triangle are:

a = 3x = 3 x 12 = 36,

b = 4x =4 x 12= 48,

c = 5x = 5 x 12= 60

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 = \(\frac{a+b+c}2\) = \(\frac{36+48+60}2\) = 72

A = \(\sqrt{72(72-36)(72-48)(72-60)}\)

A = \(\sqrt{72\times 36 \times24 \times12}\) = 864 cm²

Area of triangle = \(\frac{1}2\)(Base x Altitude)

864 = \(\frac{1}2\)(60 x Altitude)

Altitude = 28.8 cm