Question

Assertion :In any triangle $ABC$, the square of the length of the bisector $AD$ is be ${(1-\frac{a^2}{{(b+c)}^2})}^{1/2}$ Reason: In any triangle $ABC$, length of bisector $AD$ is $\frac{2bc}{(b+c)}\cos \frac{A}{2}$

• A. Both the assertion and the reason are correct and the reason is the correct explanation for the assertion
• B. Both the assertion and the reason are correct but the reason is NOT the correct explanation for the assertion
• C. The assertion is correct while the reason is incorrect
• D. The assertion is incorrect while the reason is correct

Answer 1

The correct option is A Both the assertion and the reason are correct and the reason is the correct explanation for the assertion

It is true that length of angle bisector $AD$ is given by

$\frac{2bc}{(b+c)}\cos \frac{A}{2}=\frac{2bc}{(b+c)}\left[\sqrt{\frac{1+\cos A}{2}}\right]$

$=\frac{2bc}{(b+c)}\left[\sqrt{\frac{1+\left(\frac{b^2+c^2-a^2}{2bc}\right)}{2}}\right]$

$=\frac{2bc}{(b+c)}\frac{\sqrt{{(b+c)}^2-a^2}}{4bc}=bc\sqrt{1-\frac{a^2}{{(b+c)}^2}}$

$\Rightarrow$ Assertion and reason both are correct and reason correctly explains the assertion.