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Two corners of a triangle have angles of pi / 3  and  pi / 2 . If one side of the triangle has a length of 9 , what is the longest possible perimeter of the triangle?

Jun 1, 2018

Longest possible perimeter color(red)(P = 24.59 units

Explanation:

$\hat{A} = \frac{\pi}{3} , \hat{B} = \frac{\pi}{2} , \hat{C} = \pi - \frac{\pi}{3} - \frac{\pi}{2} = \frac{\pi}{6}$

Side of length 9 should correspond to the least angle $\frac{\pi}{6}$ to get the longest perimeter.

Applying Law of Sines,

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

$a = \frac{c \sin B}{\sin} C = \frac{9 \cdot \sin \left(\frac{\pi}{6}\right)}{\sin} \left(\frac{\pi}{3}\right) = 3 \sqrt{3}$

$b = \frac{9 \sin \left(\frac{\pi}{2}\right)}{\sin} \left(\frac{\pi}{3}\right) = 6 \sqrt{3}$

Longest possible perimeter of the triangle is

$P = 9 + 6 \sqrt{3} + 3 \sqrt{3} = 24.59$ units