Two corners of a triangle have angles of $\frac{π}{3}$ $( pi )/ 3$ and $\frac{π}{4}$ $( pi ) / 4$ . If one side of the triangle has a length of $8$ $8$ , what is the longest possible perimeter of the triangle?

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Two corners of a triangle have angles of $\frac{π}{3}$ $( pi )/ 3$ and $\frac{π}{4}$ $( pi ) / 4$ . If one side of the triangle has a length of $8$ $8$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2017-10-19T21:30:50

Longest possible perimeter = 28.726

Explanation:

Three angles are $\frac{π}{3}, \frac{π}{4}, \frac{5 π}{12}$ $pi/3, pi/4, (5pi)/12$
To get longest perimeter, equate side 8 to the least angle.

$\frac{8}{sin (\frac{π}{4})} = \frac{b}{sin (\frac{π}{3})} = \frac{c}{sin (\frac{5 π}{12})}$ $8/sin (pi/4) = b/ sin (pi/3) = c/ sin ((5pi)/12)$

$b = \frac{8 \cdot sin (\frac{π}{3})}{sin (\frac{π}{4})} = \frac{8 \cdot (\frac{\sqrt{3}}{2})}{\frac{1}{\sqrt{2}}}$ $b = (8* sin(pi/3))/sin (pi/4) = (8*(sqrt3/2))/(1/sqrt2)$
$b = 8 \sqrt{\frac{3}{2}} = 9.798$ $b = 8sqrt(3/2) = 9.798$

$c = \frac{8 \cdot \frac{sin (5 π)}{12}}{sin (\frac{π}{4})} = 8 \sqrt{2} \cdot sin (\frac{5 π}{12}) = 10.928$ $c = (8*sin(5pi)/(12)) / sin (pi/4) = 8sqrt2*sin((5pi)/12) = 10.928$

Longest perimeter possible $= 8 + 9.798 + 10.928 = 28.726$ $= 8 + 9.798 + 10.928 = 28.726$

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Two corners of a triangle have angles of $\frac{π}{3}$ $( pi )/ 3$ and $\frac{π}{4}$ $( pi ) / 4$ . If one side of the triangle has a length of $8$ $8$ , what is the longest possible perimeter of the triangle?

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Explanation:

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Two corners of a triangle have angles of $\frac{π}{3}$ $( pi )/ 3$ and $\frac{π}{4}$ $( pi ) / 4$ . If one side of the triangle has a length of $8$ $8$ , what is the longest possible perimeter of the triangle?