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.

Given:

The length of the sides of a triangle are in the ratio 3:4:5 and its perimeter is 144 cm.

To do:

We have to find the area of the triangle and the height corresponding to the longest side.

 Solution:

Let the sides of the triangle be $3x, 4x$ and $5x$.

This implies,

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

$12x=144$

$x=\frac{144}{12}$

$x=12\ cm$

$3x=3(12)=36\ cm$

$4x=4(12)=48\ cm$

$5x=5(12)=60\ cm$

Therefore, the sides of the triangle are $36\ cm, 48\ cm$ and $60\ cm$.

We know that,

In a right-angled triangle, the sum of the squares of two sides is equal to the square of the hypotenuse.

$(36)^2+(48)^2=1296+2304$

$=3600$

$=(60)^2$

This implies the triangle is a right-angled triangle.

Area of the triangle$=\frac{1}{2}\times36\times48$

$=36\times24$

$=864\ cm^2$

Let the height corresponding to the longest side(hypotenuse) be $h$.

Therefore,

$\frac{1}{2}\times60\times h=864\ cm^2$

$30h=864$

$h=\frac{864}{30}$

$h=\frac{144}{5}\ cm$

The height corresponding to the longest side is $\frac{144}{5}\ cm$.