Two corners of a triangle have angles of $\frac{5 π}{8}$ $(5 pi )/ 8$ and $\frac{π}{4}$ $( pi ) / 4$ . If one side of the triangle has a length of $14$ $14$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2017-12-05T11:33:38

$A r e a o f$ $Area of$ largest possible $Delta = color (purple)(160.3294)$

Three angles are $pi/4, ((5pi)/8), (pi -( (pi/4)+ ((5pi)/8) =( pi/8)$

$a/ sin A = b / sin B = c / sin C$

To get the largest possible are, smallest angle should correspond to the side of length 14

$14 / sin (pi/8) = b / sin ((pi)/4) = c / sin ((5pi)/8)$

$b = (14*sin (pi/4)) / sin (pi / 8) = (14*(1/sqrt2)) / (0.3827) = 25.8675$

$c = (14* sin ((5pi)/8) / sin ((pi)/8) = (14 * 0.9239)/(0.3827) = 33.7983$

Semi perimeter $s = (a + b + c) / 2 = (14+ 25.8675 + 33.7983)/2 = 36.8329$

$s-a = 36.8329 -14 = 22.8329$
$s-b = 36.8329 -25.8675 = 10.9654$
$s-c = 36.8329 - 33.7983 = 3.0346$

$Area of Delta = sqrt (s (s-a) (s-b) (s-c))$

$Area of Delta = sqrt( 36.8329 * 22.8329 * 10.9654 * 3.0346)$
$Area of$ largest possible $Delta = color (purple)(160.3294)$

Two corners of a triangle have angles of (5 pi )/ 8 5π8 (5 pi )/ 8 and ( pi ) / 4 π4 ( pi ) / 4 . If one side of the triangle has a length of 14 14 14 , what is the longest possible perimeter of the triangle?