# Two corners of a triangle have angles of (5 pi ) / 12  and  ( pi ) / 8 . If one side of the triangle has a length of 4 , what is the longest possible perimeter of the triangle?

$24.459$

#### Explanation:

Let in $\setminus \Delta A B C$, $\setminus \angle A = \frac{5 \setminus \pi}{12}$, $\setminus \angle B = \setminus \frac{\pi}{8}$ hence

$\setminus \angle C = \setminus \pi - \setminus \angle A - \setminus \angle B$

$= \setminus \pi - \frac{5 \setminus \pi}{12} - \setminus \frac{\pi}{8}$

$= \frac{11 \setminus \pi}{24}$

For maximum perimeter of triangle , we must consider the given side of length $4$ is smallest i.e. side $b = 4$ is opposite to the smallest angle $\setminus \angle B = \frac{\setminus \pi}{8}$

Now, using Sine rule in $\setminus \Delta A B C$ as follows

$\setminus \frac{a}{\setminus \sin A} = \setminus \frac{b}{\setminus \sin B} = \setminus \frac{c}{\setminus \sin C}$

$\setminus \frac{a}{\setminus \sin \left(\frac{5 \setminus \pi}{12}\right)} = \setminus \frac{4}{\setminus \sin \left(\setminus \frac{\pi}{8}\right)} = \setminus \frac{c}{\setminus \sin \left(\frac{11 \setminus \pi}{24}\right)}$

$a = \setminus \frac{4 \setminus \sin \left(\frac{5 \setminus \pi}{12}\right)}{\setminus \sin \left(\setminus \frac{\pi}{8}\right)}$

$a = 10.096$ &

$c = \setminus \frac{4 \setminus \sin \left(\frac{11 \setminus \pi}{24}\right)}{\setminus \sin \left(\setminus \frac{\pi}{8}\right)}$

$c = 10.363$

hence, the maximum possible perimeter of the $\setminus \triangle A B C$ is given as

$a + b + c$

$= 10.096 + 4 + 10.363$

$= 24.459$

Jul 26, 2018

I will let you do the final calculation.

#### Explanation:

Sometimes a quick sketch helps in the understanding of the problem. That is the case hear. You only need to approximate the two given angles. It is immediately obvious (in this case) that the shortest length is AC.

So if we set this to the given permitted length of 4 then the other two are at their maximum.

The most straight forward relationship to use is the sine rule.

$\frac{A C}{\sin} \left(B\right) = \frac{A B}{\sin} \left(C\right) = \frac{B C}{\sin} \left(A\right)$ giving:

$\frac{4}{\sin} \left(\frac{\pi}{8}\right) = \frac{A B}{\sin} \left(\frac{5 \pi}{12}\right) = \frac{B C}{\sin} \left(A\right)$

We start be determining the angle A

Known: $\angle A + \angle B + \angle C = \pi \text{ radians} = 180$

$\angle A + \frac{\pi}{8} + \frac{5 \pi}{12} = \pi \text{ radians}$

$\angle A = \frac{11}{24} \pi \text{ radians" -> 82 1/2" degrees}$

This gives:

$\textcolor{b r o w n}{\frac{4}{\sin} \left(\frac{\pi}{8}\right) = \frac{A B}{\sin} \left(\frac{5 \pi}{12}\right) = \frac{B C}{\sin} \left(\frac{11 \pi}{24}\right)}$

Thus $A B = \frac{4 \sin \left(\frac{5 \pi}{12}\right)}{\sin} \left(\frac{\pi}{8}\right)$

and $B C = \frac{4 \sin \left(\frac{11 \pi}{24}\right)}{\sin} \left(\frac{\pi}{8}\right)$

Work these out and add then all up including the given length of 4