Two corners of a triangle have angles of $pi / 8$ and $pi / 3$ . If one side of the triangle has a length of $7$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-01-09T05:24:57

Longest possible perimeter of the triangle

$color(blue)(P_t = a + b + c = 12 + 27.1564 + 31.0892 = 70.2456)$

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$/_A = pi/8, /_B = pi / 3, /_C = pi - pi / 8 - pi / 3 = (13pi)/24$

To get the longest perimeter, smallest angle (/_A = pi/8) should correspond to the length $color(red)(7)$

$:. 12 / sin (pi/8) = b / sin ((pi)/3) = c / sin((13pi)/24)$

$b = (12 sin(pi/3)) / sin (pi/8) = color(red)(27.1564)$

$c = (12 sin((13pi)/24)) / sin (pi/8) = color (red)(31.0892)$

Longest possible perimeter of the triangle

$color(blue)(P_t = a + b + c = 12 + 27.1564 + 31.0892 = 70.2456)$

Two corners of a triangle have angles of pi / 8 pi / 8 and pi / 3 pi / 3 . If one side of the triangle has a length of 7 7 , what is the longest possible perimeter of the triangle?