Question

If the area of AABC is D, give the expressions that complete the equation for the measure of ZB?

Answer 1

Given:

The figure of triangle ABC.

The area of the triangle ABC is D.

$m\angle B=\sin ^{-1}(\dfrac{m}{n})$

To find:

The value of m and n in the given expression.

Solution:

Let h be the height of the triangle ABC.

Area of a triangle is:

$Area=\dfrac{1}{2}\times base\times h$

Where, b is the base and h is the height of the triangle.

$Area=\dfrac{1}{2}\times a\times h$

The area of the triangle ABC is D.

$D=\dfrac{1}{2}\times a\times h$

$2D=ah$

$\dfrac{2D}{a}=h$                  ...(i)

In a right angle triangle,

$\sin \theta =\dfrac{Perpendicular}{Hypotenuse}$

$\sin B =\dfrac{h}{c}$

$\sin B =\dfrac{1}{c}\times \dfrac{2D}{a}$              [Using (i)]

$\sin B =\dfrac{2D}{ac}$

$m\angle B =\sin ^{-1}\dfrac{2D}{ac}$            ...(ii)

We have,

$m\angle B=\sin ^{-1}(\dfrac{m}{n})$          ...(iii)

On comparing (ii) and (iii), we get

$m=2D$

$n=ac$

Therefore, the required values are $m=2D, n=ac$.