# Two corners of a triangle have angles of  (7 pi )/ 12  and  pi / 4 . If one side of the triangle has a length of  8 , what is the longest possible perimeter of the triangle?

Feb 19, 2018

Longest possible perimeter of the triangle is

color(blue)(P + a + b + c ~~ 34.7685

#### Explanation:

$\hat{A} = \frac{7 \pi}{12} , \hat{B} = \frac{\pi}{4} , s i \mathrm{de} = 8$

To find the longest possible perimeter of the triangle.

Third angle $\hat{C} = \pi - \frac{7 \pi}{12} - \frac{\pi}{4} = \frac{\pi}{6}$

To get the longest perimeter, smallest angle $\hat{C} = \frac{\pi}{6}$ should correspond to side length 8

Using sine law, $\frac{a}{\sin} A = \frac{b}{\sin} B = \frac{c}{\sin} C$

$a = \frac{c \cdot \sin A}{\sin} C = \frac{8 \cdot \sin \left(\frac{7 \pi}{12}\right)}{\sin} \left(\frac{\pi}{6}\right) = 15.4548$

$b = \frac{c \cdot \sin B}{\sin} C = \frac{8 \cdot \sin \left(\frac{\pi}{4}\right)}{\sin} \left(\frac{\pi}{6}\right) = 11.3137$

Longest possible perimeter of the triangle is

color(blue)(P + a + b + c = 15.4548 + 11.3137 + 8 = 34.7685#