Question

If the sides of a triangle are in the ratio of 5:12:13 and its perimeter is 450, find its area

Answer 1

Answer:

5+12+13=450

30=450

450/30=150

Answer 2

Answer :

Given :

Ratio of the sides - 5 : 12 : 13

Perimeter of the triangle = 450 .

Required to find :

1. Area of the triangle ?

Formula :

$\longrightarrow{\boxed{\small{Perimeter \: of \: the \: triangle \: = \: Sum \: of \: all \: it's \: side }}}$

$\longrightarrow{\boxed{\small{Heron's \: formula = \sqrt{s(s - a)(s - b)(s - c)}}}}$

Explanation :

Using the above given ratio we can find the length of the sides .

That is found by;

Equalling the sum of the ratio with perimeter .

This is because,

Perimeter = Sum of all it's sides

So, hence using the perimeter we can find the length of all three sides .

However, we have to use Heron's Formula to find the area of the triangle.

This is due to they not mentioned the length of the altitude of the triangle .

For Using Heron's Formula . we should find the semi perimeter.

Semi perimeter is the half of the perimeter .

Now, let's crack the solution .

Answer :

Ratio : 5 : 12 : 13

Perimeter of the triangle = 450

So, let's consider the sides be ; 5x , 12x , 13x

( where 'x' is any positive integer )

According to problem;

5x + 12x + 13x = 450

30x = 450

$\Rightarrow{x = \frac{450}{30}}$

$\implies{\boxed{x = 15}}$

Hence, the sides are ;

5x = 5(15) = 75 units

12x = 12(15) = 180 units

13x = 13(15) = 195 units

Now, let's find the area of the triangle using Heron's formula ;

Formula is ;

$\longrightarrow{\boxed{\small{Heron's \: formula = \sqrt{s(s - a)(s - b)(s - c)}}}}$

here,

s = semi perimeter .

a,b,c are the three sides of the triangle .

Semi perimeter is half of the perimeter

So,

$\longrightarrow{semi-perimeter = \frac{450}{2}}$

$\Rightarrow{S = \cancel{ \frac{450}{2}}}$

$\boxed{\implies{S = 225}}$

Now, substitute the required values in the formula .

We get,

$\longrightarrow{ Area = \sqrt{225(225-75)(225-180)(225-195)}}$

$\longrightarrow{ Area = \sqrt{225(150)(45)(30)}}$

$\Rightarrow{ Area = \sqrt{(5 \times 45)(5 \times 30)(45)(30)}}$

$\Rightarrow{ Area = \sqrt{(5 \times 5)(30 \times 30)(45 \times 45)}}$

$\Rightarrow{ Area = \sqrt{{5}^{2} \times {30}^{2} \times {45}^{2}}}$

$\Rightarrow{ Area = 5 \times 30 \times 45}$

$\implies{Area = 6,750 \:square \:units}$

$\boxed{\boxed{\therefore{Area\:of\:the\: triangle\:=\:6,750\: square\: units}}}$

✓ Hence Solved .