Question

find the area of the triangle whose two sides are 18cm and 10 cm and perimeter is 42 cm using heron's formula class 9​

Answer 1

Given :

Sides of the triangle = 18cm & 10cm .

Perimeter of the triangle = 42cm .

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Required to find :

• Area of the triangle

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Mentioned Condition :

Using Heron's Formula

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Formulae Used :

Heron's Formula :-

$\small{\boxed{\tt{Area \: of \: the \: triangle \: = \sqrt{s(s - a)(s - b)(s - c)} }}}$

Semi-perimeter :-

$\boxed{\tt{semi - perimeter = \dfrac{a + b + c}{2} }}$

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Solution :

Given sides :-

18cm & 10 cm

The measurement of the third side is not given .

So,

Let the third side be 'x'

Perimeter of the triangle = 42 cm

According to problem

Sum of all sides of the triangle = Perimeter of the triangle

So,

➦ 18 + 10 + x = 42

➦ 28 + x = 42

➦ x = 42 - 28

➦ x = 14 cm

Length of third side = 14 cm

Now, let's find the semi-perimeter of the triangle ,

By using the formula

$\rightarrow{\tt{ Semi-perimeter = \dfrac{a + b + c}{2}}}$

Hence,

$\Rightarrow{\tt{ Semi-perimeter = \dfrac{18+10+14}{2}}}$

$\Rightarrow{\tt{Semi-perimeter = \dfrac{42}{2}}}$

$\implies{\tt{ S = 21 cm }}$

Heron's Formula :-

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

Using Heron's Formula let's find the area of the triangle .

Here, substitute the respective values ;

$\tt{Area\;of\; triangle = \sqrt{ 21 (21 - 18)(21-10)(21-14)}}$

$\tt{Area\:of\: triangle = \sqrt{21 ( 3)(11)(7)}}$

$\tt{ Area = \sqrt{ 3 \times 7 \times 3 \times 11 \times 7 }}$

$\tt{ Area = \sqrt{ {3}^{2} \times {7}^{2} \times 11}}$

$\tt{Area = \sqrt{ {3}^{2}} \times \sqrt{ {7}^{2}} \times \sqrt{11}}$

Here, squares and square roots get cancelled except in the case of 11 .

$\tt{ Area = \sqrt{ 3 \times 7 \sqrt{11}}}$

$\tt{ Area = 21 \sqrt{11}}$

As we know that ;

$\green{\longrightarrow{ \sqrt{11} = 3.316(approximately)}}$

Hence,

$\tt{ Area = 21 \times 3.316 }$

$\implies{\tt{ Area = 69.636 {cm}^{2} (approximately)}}$

Answer :

$\boxed{\tt{\therefore{Area\:of\: triangle= 69.636 {cm}^{2}}}}$

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Explanation :

Heron's Formula :-

$\small{\boxed{\tt{Area \: of \: the \: triangle \: = \sqrt{s(s - a)(s - b)(s - c)} }}}$

Here,

S represents Semi-perimeter .

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

Similarly ,

Semi-perimeter is the half of the perimeter .

It is found by the formula ;

$\boxed{\tt{semi - perimeter = \dfrac{a + b + c}{2} }}$

The Heron's Formula is used when the measurement of the height is not given .

This Formula is credited to Heron of Alexandria who is a Greek mathematician and engineer during the 10 - 70 AD .

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