Two corners of a triangle have angles of $\frac{5 π}{8}$ $(5 pi )/ 8$ and $\frac{π}{12}$ $( pi ) / 12$ . If one side of the triangle has a length of $18$ $18$ , what is the longest possible perimeter of the triangle?

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Two corners of a triangle have angles of $\frac{5 π}{8}$ $(5 pi )/ 8$ and $\frac{π}{12}$ $( pi ) / 12$ . If one side of the triangle has a length of $18$ $18$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2016-10-22T14:50:32

Longest possible perimeter is $137.434$ $137.434$

Explanation:

As two angles are $\frac{5 π}{8}$ $(5pi)/8$ and $\frac{π}{12}$ $pi/12$ , third angle is

$π - \frac{5 π}{8} - \frac{π}{12} = \frac{24 π}{24} - \frac{15 π}{24} - \frac{2 π}{24} = \frac{7 π}{24}$ $pi-(5pi)/8-pi/12=(24pi)/24-(15pi)/24-(2pi)/24=(7pi)/24$

the smallest of these angles is $\frac{π}{12}$ $pi/12$

Hence, for longest possible perimeter of the triangle, the side with length $18$ $18$ , will be opposite the angle $\frac{π}{12}$ $pi/12$ .

Now for other two sides, say $b$ $b$ and $c$ $c$ , we can use sine formula, and using it

$\frac{18}{sin (\frac{π}{12})} = \frac{b}{sin (\frac{5 π}{8})} = \frac{c}{sin (\frac{7 π}{24})}$ $18/sin(pi/12)=b/sin((5pi)/8)=c/sin((7pi)/24)$

or $\frac{18}{0.2588} = \frac{b}{0.9239} = \frac{c}{0.7933}$ $18/0.2588=b/0.9239=c/0.7933$

therefore $b = \frac{18 \times 0.9239}{0.2588} = 64.259$ $b=(18xx0.9239)/0.2588=64.259$

and $c = \frac{18 \times 0.7933}{0.2588} = 55.175$ $c=(18xx0.7933)/0.2588=55.175$

and perimeter is $64.259 + 55.175 + 18 = 137.434$ $64.259+55.175+18=137.434$

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Two corners of a triangle have angles of $\frac{5 π}{8}$ $(5 pi )/ 8$ and $\frac{π}{12}$ $( pi ) / 12$ . If one side of the triangle has a length of $18$ $18$ , what is the longest possible perimeter of the triangle?

1 Answer

Explanation:

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Two corners of a triangle have angles of (5 pi )/ 8 5π8 (5 pi )/ 8 and ( pi ) / 12 π12 ( pi ) / 12 . If one side of the triangle has a length of 18 18 18 , what is the longest possible perimeter of the triangle?

1 Answer

Explanation:

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Two corners of a triangle have angles of $\frac{5 π}{8}$ $(5 pi )/ 8$ and $\frac{π}{12}$ $( pi ) / 12$ . If one side of the triangle has a length of $18$ $18$ , what is the longest possible perimeter of the triangle?