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 triangle and the height corresponding to the longest side

Answer 1

Let the sides of triangles are $3x,4x$ and $5x.$

And perimeter of the triangle is 144cm.

So , $3x+4x+5x=144$

$12x=144$

$x=12$

so, the sides are triangle are $3\left(12\right)=36cm$, $4\left(12\right)=48cm$ , $5\left(12\right)=60cm$

so , the longest side is 60 cm.

Area of triangle $=$$\sqrt{s(s-a)(s-b)(s-c)}$

here , $s=72$ and $a=36$, $b=48$ $c=60$

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

$=$ $\sqrt{72\left(36\right)\left(24\right)\left(12\right)}$

$=$ $\sqrt{746496}$

$=$ $864$

So, the area of triangle is 864 $c{m}^2$ and the height corresponding to longest side $h=\frac{2\times 864}{60}=28.8cm$