The perimeter of a triangular field is 300 cm and its sides are in the ratio 5:12:13 .Find the length of the perpendicular from the opposite vertex to the longest side.

Given:

The lengths of the sides of a triangle are in the ratio $5:12:13$ and its perimeter is 300 cm.

To do:

We have to find the length of the perpendicular from the opposite vertex to the longest side. 

 Solution:

Let the sides of the triangle be $5x, 12x$ and $13x$.

This implies,

$5x+12x+13x=300$

$30x=300$

$x=\frac{300}{30}$

$x=10\ cm$

$5x=5(10)=50\ cm$

$12x=12(10)=120\ cm$

$13x=13(10)=130\ cm$

Therefore, the sides of the triangle are $50\ cm, 120\ cm$ and $130\ cm$.

We know that,

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

$(50)^2+(120)^2=2500+14400$

$=16900$

$=(130)^2$

This implies the triangle is a right-angled triangle.

Area of the triangle$=\frac{1}{2}\times50\times120$

$=25\times120$

$=3000\ cm^2$

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

Therefore,

$\frac{1}{2}\times130\times h=3000\ cm^2$

$65h=3000$

$h=\frac{3000}{65}$

$h=46.15\ cm$

The height corresponding to the longest side is $46.15\ cm$. 

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Updated on: 10-Oct-2022

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