Question

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

Answer 1

Perimeter = 144 cm and ratio of sides = 3 : 4 : 5
Sum of ratio terms = (3 + 4 + 5) = 12.
Let the lengths of the sides be a, b and c respectively.

Then, $a=(144\times \frac{3}{12})cm=36cm,b=(144\times \frac{4}{12})cm=48cm$ and $c=(144\times \frac{4}{12})cm=60cm$
$\therefore s=\frac{1}{2}(a+b+c)=\frac{1}{2}(36+48+60)cm=72cm$.
By Heron's formula, the area of the triangle is given by
$\Delta =\sqrt{s(s-a)(s-b)(s-c)}$
$=\sqrt{72\times 36\times 24\times 12}c{m}^2$
$=\sqrt{36\times 36\times 24\times 24}c{m}^2$
$=(36\times 24)c{m}^2=864c{m}^2$
Hence, the area of the given triagnel is $864c{m}^2$.
Let base $=$ longest side = $60cm$ and the correspoindig height $=hcm$
Then, area $=(\frac{1}{2}\times \text{base}\times \text{height})\text{sq units}$
$=(\frac{1}{2}\times 60\times h)c{m}^2=\left(30h\right)c{m}^2$
$\therefore 30h=864\Rightarrow h=\left(\frac{864}{30}\right)=28.8cm$
Hence, the height corresponding to the longest side is $28.8cm$