Question

The longest side of a right angled triangle is 125 m and one of the remaining two sides is 100 m. Find its area using Heron's formula.

Answer 1

Given that,

The longest side of a right angled triangle is $125m$.

One of the other two sides is $100m$

To find out,

The area of the triangle using Heron's formula.

Let the given triangle be $ABC$, right angled at $B$.

Hence, $AC=125m$ and let $BC=100m$

Hence, by Pythagoras theorem,

$A{C}^2=A{B}^2+B{C}^2$

$\Rightarrow A{B}^2={125}^2-{100}^2$

$\Rightarrow A{B}^2=15625-10000$

$\Rightarrow A{B}^2=5625$

Hence, $AB=75m$

We know that, area of a triangle is $\sqrt{s(s-a)(s-b)(s-c)}$

where, $a,b,c$ are the three sides and $s$ is the semi-perimeter of the triangle.

Thus, $s=\frac{a+b+c}{2}$

Here, $s=\frac{75+125+100}{2}$

$=150m$

Hence, area $=\sqrt{150\left(25\right)\left(50\right)\left(75\right)}$

$=\sqrt{(75\times 2)\left(25\right)(25\times 2)\left(75\right)}$

$=\sqrt{(75\times 75)(25\times 25)(2\times 2)}$

$=75\times 25\times 2$

$=75\times 50$

$=3750m^2$

Hence, area of the triangle is $3750m^2$