Two corners of a triangle have angles of $\frac{5 π}{12}$ $(5 pi )/ 12$ and $\frac{π}{3}$ $( pi ) / 3$ . If one side of the triangle has a length of $15$ $15$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-02-19T05:21:52

Longest possible perimeter

$p = a + b + c \approx 53.86$ $p = a + b + c ~~ color (green)(53.86$

To longest possible perimeter of the triangle.

Given : $ˆ A = \frac{5 π}{12}, ˆ B = \frac{π}{3}$ $hatA = (5pi)/12, hatB = pi/3$ , one $s i d e = 15$ $side = 15$

Third angle $ˆ C = π - \frac{5 π}{12} - \frac{π}{3} = \frac{π}{4}$ $hatC = pi - (5pi)/12 - pi/3 = pi/4$

To get the longest perimeter, side 15 should correspond to the smallest angle $ˆ C = \frac{π}{4}$ $hatC = pi/4$

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

$\frac{a}{sin (5 π)} / 12 = \frac{b}{sin (\frac{π}{3})} = \frac{15}{sin (\frac{π}{4})}$ $a / sin (5pi)/12 = b / sin(pi/3) = 15 / sin (pi/4)$

$a = \frac{15 \cdot sin (\frac{5 π}{12})}{sin (\frac{π}{4})} \approx 20.49$ $a = (15 * sin ((5pi)/12)) / sin (pi/4) ~~ 20.49$

$b = \frac{15 \cdot \frac{sin (π)}{3}}{sin (\frac{π}{4})} \approx 18.37$ $b = (15 * sin (pi)/3) / sin (pi/4) ~~ 18.37$

Longest possible perimeter

$p = a + b + c = 20.49 + 18.37 + 15 = 53.86$ $p = a + b + c = 20.49 + 18.37 + 15 = color (green)(53.86$

Two corners of a triangle have angles of 5π12 (5 pi )/ 12 and π3 ( pi ) / 3 . If one side of the triangle has a length of 15 15 , what is the longest possible perimeter of the triangle?