Two corners of a triangle have angles of $pi / 8$ and $pi / 6$ . If one side of the triangle has a length of $4$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2016-06-02T06:12:37

Longest possible perimeter is $17.519$ .

As two angles are $pi/8$ and $pi/6$ , third angle is $pi-pi/8-pi/6=(24pi-3pi-4pi)/24-(17pi)/24$ .

For longest perimeter side of length $4$ , say $a$ , has to be opposite smallest angle $pi/8$ and then using sine formula other two sides will be

$4/(sin(pi/8))=b/(sin(pi/6))=c/(sin((17pi)/24))$

Hence $b=(4sin(pi/6))/(sin(pi/8))=(4xx0.5)/0.3827=5.226$

and $c=(4xxsin((17pi)/24))/(sin(pi/8))=(4xx0.7934)/0.3827=8.293$

Hence longest perimeter is $4+5.226+8.293=17.519$ .

Two corners of a triangle have angles of pi / 8 pi / 8 and pi / 6 pi / 6 . If one side of the triangle has a length of 4 4 , what is the longest possible perimeter of the triangle?