Question

In $\Delta$ $ABC$, if $a$,$b$ and $A$ are given then two triangles can be formed with the third side having lengths $c_1$ and $c_2$. The sum of areas of these two triangles is?

• A. $\frac{1}{2}{b}^2\cos 2A$
• B. $\frac{1}{2}{b}^2\sin 2A$
• C. $b^2{\cos }^{2}A$
• D. $b^2\cos 2A$

Answer 1

The correct option is B $\frac{1}{2}{b}^2\sin 2A$

$a^2=b^2+c^2-2bc\cos A\left(cosineLaw\right)$

$c^2-2bc\cos A+(b^2-a^2)=0c_1+c_2=2b\cos A,c_1{c}_2=b^2-a^2$

Sum of triangle $=\frac{1}{2}b{c}_1\sin A+\frac{1}{2}b{c}_2\sin A$

$=\frac{1}{2}b\sin A(c_1+c_2)$

$=\frac{1}{2}{b}^2\sin A\left(2\cos A\right)$

$=\frac{1}{2}{b}^2\sin 2A$