Question

In $\Delta ABC,a,b,A$ are given and $c_1,c_2$ are two values of the third side $c$. The sum of the areas of the two triangles with sides $a,b$, $c_1$ and $a,b$, $c_2$ is

• A. $\left(1/2\right)b^2\sin 2A$
• B. $\left(1/2\right)a^2\sin 2A$
• C. $b^2\sin 2A$
• D. none of these

Answer 1

Answer: (A)

$\left(1/2\right)b^2\sin 2A$

From cosine rule we have

$\Rightarrow c^2-2bc\cos A+b^2-a^2=0$

let $c_1$ and $c_2$ be the roots of the above equation.

The sum of the roots $=$ $c_1+c_2=2b\cos A$

Now, sum of area of two triangle=$\Delta ={\Delta }_1+{\Delta }_2=\frac{b{c}_1\sin A+b{c}_2\sin A}{2}=\frac{b(c_1+c_2)\sin A}{2}$

$\Rightarrow △=b^2\cos A\sin A=\frac{b^2\sin 2A}{2}$