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# The sides of a triangle are 21cm,17cm and 10cm.find its area.

Last updated date: 20th Jun 2024
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Hint: In order to solve this question, we need to calculate the area of the scalene triangle. The area of a scalene triangle with base 'b' and height 'h' is given by 1/2 bh. If you know the lengths of all three sides, you can calculate the area using Heron's Formula without having to find the height. In this question, we can get our final result only by substituting the values of the sides of the triangle in the heron's formula to get the area.
Area of scalene triangle is : $\sqrt{S\left( S-a \right)\left( S-b \right)\left( S-c \right).}$

Complete step by step solution: The sides of a triangle are given as 21cm, 17cm and 10cm.
Here we see that the three sides are different, so it must be a scalene triangle.
The area of the scalene triangle is given is:
$\text{Area}\ \text{=}\sqrt{\text{S}\left( \text{S-a}\left( \text{S-b} \right)\left( \text{S-b} \right) \right)}$
Where, ‘S’ is the semi-perimeter and, a, b. c are the sides of the triangle.
$\text{So,}\ \text{a}\ \text{=}\ \text{21cm ;}\ \text{b}\ \text{=}\ \text{17cm}\ \text{and}\ \text{c}\ \text{=}\ \text{10cm}\text{.}$
$\text{and,}\ \text{S=}\dfrac{a+b+c}{2}$
$\Rightarrow S=\dfrac{21cm+17cm+10cm}{2}$
$\Rightarrow \text{S=}\dfrac{48cm}{2}$
$\Rightarrow \text{S}=\ 24cm$
So, Area $=\sqrt{S\left( s-a \right)\left( s-b \right)\left( s-c \right)}$
$\Rightarrow \text{Area}\ \text{=}\ \sqrt{24cm\left( 24-21cm \right)}\left( 24-17cm \right)cm\left( 24-10 \right)cm$
$\Rightarrow \ \text{Area}\ \text{=}\sqrt{24cm\times 3cm\times 7cm\times 14cm}$
$\Rightarrow \ \text{Area=}\sqrt{72c{{m}^{2}}\times 98c{{m}^{2}}}$
$\Rightarrow \ \text{Area=}\sqrt{7056c{{m}^{4}}}$
$\Rightarrow \text{Area=}\;84\;\text{cm}^{2}$

$\therefore$ Area of the triangle with sides 21cm, 17cm and 10cm is $84\;\text{cm}^{2}$

Note: For finding out the area of a scalene triangle, we need to do the following measurements:-
a) The length of one side of the triangle and the perpendicular distance of that particular side to the opposite angle.
b) The lengths of all three sides of the triangle.
So, Area of scalene triangle is given by: - and here, $\text{S}=\,\dfrac{(\text{a}+\text{b}+\text{c})}{2}$where a, b, c are sides of triangle and 'S' is the semi-perimeter of triangle having sides a,b,c.