Question

Which of the following statements is true?

I: ln a $\Delta ABC$ if $a=R\tan A$, then $b^2+c^2=bc+a^2$
II: In a $\Delta ABC$ if $a\sin A=c\cos B+b\cos C$, then the triangle is right angled.

• A. only I
• B. only II
• C. both I and II
• D. neither I nor II

Answer 1

The correct option is C both I and II

According to sine rule,

it is given that $\frac{b^2+c^2-a^2}{bc}=1$

On dividing both sides by 2 we get,

$\Rightarrow \frac{b^2+c^2-a^2}{2bc}=\frac{1}{2}\Rightarrow \cos A=\frac{1}{2}$ .........(1)

Now,

$\frac{a}{\sin A}=2R\Rightarrow a=2R\sin A\Rightarrow a=\frac{R\sin A}{\frac{1}{2}}$

from equation (1) we can write $a=\frac{R\sin A}{\cos A}=R\tan A$

so, $I$ is correct

Now, from the projection Formula

we can see that

$a=c\cos B+b\cos C$

and it is given that $a\sin A=c\cos B+b\cos C$ which is only true when $\sin A=1\Rightarrow A={90}^o$

Since, one of the angle of triangle is right angle then the triangle is Right Angled triangle.

Hence, $II$ is correct

Therefore Answer is $C$