Question

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 1

Ratio in the sides of a triangle= 3 : 4 : 5
Perimeter = 144 cm
Sum of ratios= 3+4+5=12
$\therefore$ First side = $\frac{144}{12}\times 3=36cm$
Second side = $\frac{144}{12}\times 4=48cm$
Third side = $\frac{144}{12}\times 5=60cm$
$\therefore s=\frac{Perimeter}{2}=\frac{144}{2}=72$
$\therefore Area=\sqrt{s(s-a)(s-b)(s-c)}=\sqrt{72(72-36)(72-48)(72-60)}=\sqrt{72\times 36\times 24\times 12}\sqrt{12\times 3\times 2\times 12\times 3\times 12\times 2\times 12}=12\times 12\times 3\times 2=864c{m}^2$
Length of altitude on the longest side corresponds to 60 cm which is equal to
$\frac{Area\times 2}{Base}=\frac{864\times 2}{60}=\frac{1728}{60}$
$=\frac{288}{10}=28.8cm$