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

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

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Answer 1 · 2016-06-02T06:12:37

Longest possible perimeter is $17.519$ $17.519$ .

Explanation:

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

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

$\frac{4}{sin (\frac{π}{8})} = \frac{b}{sin (\frac{π}{6})} = \frac{c}{sin (\frac{17 π}{24})}$ $4/(sin(pi/8))=b/(sin(pi/6))=c/(sin((17pi)/24))$

Hence $b = \frac{4 sin (\frac{π}{6})}{sin (\frac{π}{8})} = \frac{4 \times 0.5}{0.3827} = 5.226$ $b=(4sin(pi/6))/(sin(pi/8))=(4xx0.5)/0.3827=5.226$

and $c = \frac{4 \times sin (\frac{17 π}{24})}{sin (\frac{π}{8})} = \frac{4 \times 0.7934}{0.3827} = 8.293$ $c=(4xxsin((17pi)/24))/(sin(pi/8))=(4xx0.7934)/0.3827=8.293$

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

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

1 Answer

Explanation:

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