Question

Find the area of the right angled triangle whose hypotenuse is $13cm$ and the base is $12cm$.

Answer 1

Hint: Here we are given the values of the hypotenuse and the base of the right-angled triangle. We need to calculate the area of the right-angled triangle. First, we shall calculate the height of the triangle, and then we need to substitute the obtained values in the formula of the area of the triangle.

Formula used:
a) ${c^2} = {a^2} + {b^2}$
Here c is the hypotenuse that is the largest side of the right-angle triangle, and a is the base that is the bottom side and b is the height that is the altitude of the right-angled triangle.
b) \[Area{\text{ }}of{\text{ }}the{\text{ }}triangle = \dfrac{1}{2} \times base \times height\]

Complete step by step solution:

It is given that the hypotenuse is $13cm$ and the base is $12cm$ and we are asked to calculate the area of the right-angled triangle.
To find the area of the triangle, we need two quantities, the base, and the height.
But we are not provided the value of the height.
By Pythagoras theorem, we know that the square of the hypotenuse is equal to the sum of the squares of the base and the height.
Since we are given hypotenuse and base, we need to just apply the formula ${c^2} = {a^2} + {b^2}$to obtain the height of the triangle.
Thus, we get ${13^2} = {12^2} + heigh{t^2}$
$ \Rightarrow 169 = 144 + heigh{t^2}$
$ \Rightarrow heigh{t^2} = 169 - 144$
$ \Rightarrow heigh{t^2} = 25$
$ \Rightarrow height = 5cm$
Thus we obtained the height of the right-angle triangle.
Now, we need to find the required area of the right-angled triangle.
We know that \[Area{\text{ }}of{\text{ }}the{\text{ }}triangle = \dfrac{1}{2} \times base \times height\]
Thus, we have \[Area{\text{ }}of{\text{ }}the{\text{ }}triangle = \dfrac{1}{2} \times 12 \times 5\]
$ \Rightarrow 6 \times 5$
$ \Rightarrow 30$
Therefore, the required area of the right angle triangle is $30c{m^2}$ .

Note:
 If we are given the values of the base and the height of the right-angled triangle directly, we just need to substitute the given values in the formula of the area of the triangle. Suppose we are not given the value of the base of the right-angled triangle, we need to do the same steps we did here.