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 $9$ $9$ , 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 $9$ $9$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2016-05-24T09:55:28

Perimeter is $32.314$ $32.314$

Explanation:

As two angles of a triangle are $\frac{π}{3}$ $pi/3$ and $\frac{π}{4}$ $pi/4$ , the third angle is

$π - \frac{π}{3} - \frac{π}{4} = (12 - 4 - 3) \frac{π}{12} = \frac{5 π}{12}$ $pi-pi/3-pi/4=(12-4-3)pi/12=(5pi)/12$

Now for the longest possible perimeter, the given side say $B C$ $BC$ , should be the smallest angle $\frac{π}{4}$ $pi/4$ , let this be $∠ A$ $/_A$ . Now using sine formula

$\frac{9}{sin (\frac{π}{4})} = \frac{A B}{sin (\frac{π}{3})} = \frac{A C}{sin (\frac{5 π}{12})}$ $9/sin(pi/4)=(AB)/sin(pi/3)=(AC)/sin((5pi)/12)$

Hence $A B = 9 \times \frac{sin (\frac{π}{3})}{sin (\frac{π}{4})} = 9 \times \frac{\frac{\sqrt{3}}{2}}{\frac{\sqrt{2}}{2}} = 9 \times \frac{1.732}{1.414} = 11.02$ $AB=9xxsin(pi/3)/sin(pi/4)=9xx(sqrt3/2)/(sqrt2/2)=9xx1.732/1.414=11.02$

and $A C = 9 \times \frac{sin (\frac{5 π}{12})}{sin (\frac{π}{4})} = 9 \times \frac{0.9659}{\frac{1.4142}{2}} = 12.294$ $AC=9xxsin((5pi)/12)/sin(pi/4)=9xx0.9659/(1.4142/2)=12.294$

Hence, perimeter is $9 + 11.02 + 12.294 = 32.314$ $9+11.02+12.294=32.314$

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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 $9$ $9$ , what is the longest possible perimeter of the triangle?

1 Answer

Explanation:

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Two corners of a triangle have angles of π3 ( pi )/ 3 and π4 ( pi ) / 4 . If one side of the triangle has a length of 9 9 , 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 $9$ $9$ , what is the longest possible perimeter of the triangle?