# Finding more details about a triangle using the given details.

In the triangle $ABC$ we have

$\tan{\frac{A}{2}}=\frac{1}{3}$

$b+c=3a$

Specify which of the following answers is correct:

$a) m(\angle B)=\frac{\pi}{2}$ or $m(\angle C)=\frac{\pi}{2}$

$b) m(\angle A)=m(\angle B)$

$c) m(\angle A)=\frac{\pi}{2}$

$d) m(\angle B)=\frac{\pi}{4}$ or $m(\angle C)=\frac{\pi}{4}$

$e) m(\angle A)=m(\angle C)$

$f) m(\angle A)=\frac{\pi}{3}$

I'm lost here. I don't know how I can use $\tan{\frac{A}{2}}=\frac{1}{3}$ so I get to an answer. Can someone give me a solution? I have many exercises involving the $\tan{\frac{A}{2}}$ function and I can't continue.

Thank you very much!

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What if the question said $\tan x = \frac{1}{3}$? What are the possible values for $x$? – Code-Guru Jul 18 '12 at 19:29
I'm confused about that. I don't know how to use $\tan\frac{A}{2}$. – Grozav Alex Ioan Jul 18 '12 at 19:31
What I'm saying is ignore the $\frac{A}{2}$ and just look at the $\tan$ function. What angles have a tangent value of $\frac{1}{3}$? – Code-Guru Jul 18 '12 at 19:33
Another approach is to evaluate $\tan x$ where $x$ is any of the values given in the choices involving the angle $A$. For example, what is $\tan \frac{\pi}{2}$? – Code-Guru Jul 18 '12 at 19:36

I will assume that in this multiple choice question only one of the answers can be correct. If we do not assume that, there is additional work to do.

It is handy but not necessary to know the formula $$\tan(x+y)=\frac{\tan x+\tan y}{1-\tan x\tan y}.$$ Putting $x=y=\frac{A}{2}$, after a while we get $\tan A =\frac{3}{4}$.

This is what we get with the good old $3$-$4$-$5$ right triangle with sides $3k$, $4k$, $5k$. And by a miracle we have in that case that $4k+5k=3(3k)$. So the second condition $b+c=3a$ is met.

So a right angle at $B$ or $C$ will do the job. If there is a unique answer, that answer is a).

Remark: We could get there with a calculator. Use it to find (approximately) $A/2$, and then $A$. Not very informative. But then we may get the lucky idea of computing $\tan A$, (or $\sin A$, or $\cos A$.) The calculator gives $0.75$, (or $0.6$, or $0.8$) and then we recognize the familiar triangle.

If we are not allowed to assume a unique answer to the multiple choice question, we need to rule out the other possibilities. Some are very quick to rule out. The possibilities b), d), and e) take a little longer.

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Thank you very much! Helped me solve the rest of the problems too. Didn't think about plugging in $x=y=\frac{A}{2}$. – Grozav Alex Ioan Jul 19 '12 at 6:34
@GrozavAlexIoan: It is good to hear that the idea helped you solve other problems. – André Nicolas Jul 19 '12 at 6:40