Question

Calculate the area of triangle whose sides are 18cm,24cm and 30cm in length.Hence, find the length of the altitude corresponding to the smallest side. Solution

Answer 1

Heya !!

$\text{Let's find out the area first.}\\\\\text{Area of a triangle by Heron's formula = \sqrt{s(s-a)(s-b)(s-c)} ,}\\\\\text{where, a, b and c are the three sides of the triangle and s = \dfrac{a+b+c}{2}= \dfrac{18+24+30}{2} = 36 }\\\\\text{Therefore, area of \triangle = \sqrt{36(36-18)(36-24)(36-30)} }\\\\= \sqrt{36 \times 18 \times 12 \times 6} = \large \underline{\boxed{\boxed{216 \: \: cm^2}}}$

Now,

$\text{Smallest side = 18 cm}\\\\\text{Let the height corresponding to it be \alpha .}\\\\\text{Area of \triangle = \dfrac{1}{2} \times Base \times Height }\\\\= \dfrac{1}{2} \times 18 \times \alpha = 9 \alpha\\\\\text{But, area of triangle was 216 cm^2 .}\\\\\therefore 9 \alpha = 216\\\\\implies \alpha = \large \underline{\boxed{\boxed{24 \: \: cm}}}$

Hope it helps !!