Two corners of a triangle have angles of $\frac{π}{3}$ $pi / 3$ and $\frac{π}{2}$ $pi / 2$ . If one side of the triangle has a length of $7$ $7$ , 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 $\frac{π}{2}$ $pi / 2$ . If one side of the triangle has a length of $7$ $7$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2016-05-12T09:01:19

Longest possible perimeter is $33.124$ $33.124$ .

Explanation:

As two angles are $\frac{π}{2}$ $pi/2$ and $\frac{π}{3}$ $pi/3$ , the third angle is $π - \frac{π}{2} - \frac{π}{3} = \frac{π}{6}$ $pi-pi/2-pi/3=pi/6$ .

This is the least angle and hence side opposite this is smallest.

As we have to find longest possible perimeter, whose one side is $7$ $7$ , this side must be opposite the smallest angle i.e. $\frac{π}{6}$ $pi/6$ . Let other two sides be $a$ $a$ and $b$ $b$ .

Hence using sine formula $\frac{7}{sin (\frac{π}{6})} = \frac{a}{sin (\frac{π}{2})} = \frac{b}{sin (\frac{π}{3})}$ $7/sin(pi/6)=a/sin(pi/2)=b/sin(pi/3)$

or $\frac{7}{\frac{1}{2}} = \frac{a}{1} = \frac{b}{\frac{\sqrt{3}}{2}}$ $7/(1/2)=a/1=b/(sqrt3/2)$ or $14 = a = 2 \frac{b}{\sqrt{3}}$ $14=a=2b/sqrt3$

Hence $a = 14$ $a=14$ and $b = 14 \times \frac{\sqrt{3}}{2} = 7 \times 1.732 = 12.124$ $b=14xxsqrt3/2=7xx1.732=12.124$

Hence, longest possible perimeter is $7 + 14 + 12.124 = 33.124$ $7+14+12.124=33.124$

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Two corners of a triangle have angles of $\frac{π}{3}$ $pi / 3$ and $\frac{π}{2}$ $pi / 2$ . If one side of the triangle has a length of $7$ $7$ , what is the longest possible perimeter of the triangle?

1 Answer

Explanation:

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Two corners of a triangle have angles of $\frac{π}{3}$ $pi / 3$ and $\frac{π}{2}$ $pi / 2$ . If one side of the triangle has a length of $7$ $7$ , what is the longest possible perimeter of the triangle?