Two corners of a triangle have angles of $\frac{π}{3}$ $pi / 3$ and $\frac{π}{6}$ $pi / 6$ . If one side of the triangle has a length of $7$ $7$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-06-02T09:00:49

Longest possible Perimeter $P = 33.12$ $color(brown)(P = 33.12$

$ˆ A = \frac{π}{3}, ˆ B = \frac{π}{6}, ˆ C = \frac{π}{2}$ $hat A = pi/3, hat B = pi/6, hat C = pi/2$

To get the longest perimeter, side 7 should correspond to the least angle $ˆ B$ $hat B$

$a = \frac{b sin A}{sin B} = \frac{7 sin (\frac{π}{3})}{sin (\frac{π}{6})} = 12.12$ $a = ( b sin A) / sin B = (7 sin (pi/3)) / sin (pi/6) = 12.12$

$c = \frac{b \cdot sin C}{sin B} = \frac{7 sin (\frac{π}{2})}{sin (\frac{π}{6})} = 14$ $c = ( b * sin C) / sin B = (7 sin (pi/2)) / sin (pi/6) = 14$

Perimeter of the triangle $P = 7 + 12.12 + 14 = 33.12$ $color(brown)(P = 7 + 12.12 + 14 = 33.12$

Two corners of a triangle have angles of pi / 3 π3pi / 3 and pi / 6 π6 pi / 6 . If one side of the triangle has a length of 7 77 , what is the longest possible perimeter of the triangle?