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 $9$ $9$ , 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 $\frac{π}{2}$ $pi / 2$ . 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 · 2018-06-01T15:17:51

Longest possible perimeter $P = 24.59$ $color(red)(P = 24.59$ units

Explanation:

$ˆ A = \frac{π}{3}, ˆ B = \frac{π}{2}, ˆ C = π - \frac{π}{3} - \frac{π}{2} = \frac{π}{6}$ $hat A = pi/3, hat B = pi/2, hat C = pi - pi/3 - pi /2 = pi/6$

Side of length 9 should correspond to the least angle $\frac{π}{6}$ $pi/6$ to get the longest perimeter.

Applying Law of Sines,

$\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}$ $a / sin A = b / sin B = c / sin C$

$a = \frac{c sin B}{sin C} = \frac{9 \cdot sin (\frac{π}{6})}{sin (\frac{π}{3})} = 3 \sqrt{3}$ $a = (c sin B) / sin C = (9 * sin (pi/6)) / sin (pi/3) = 3 sqrt3$

$b = \frac{9 sin (\frac{π}{2})}{sin (\frac{π}{3})} = 6 \sqrt{3}$ $b = (9 sin(pi/2)) / sin (pi/3) = 6sqrt3$

Longest possible perimeter of the triangle is

$P = 9 + 6 \sqrt{3} + 3 \sqrt{3} = 24.59$ $P = 9 + 6sqrt3 + 3sqrt3 = 24.59$ units

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

1 Answer

Explanation:

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Two corners of a triangle have angles of $\frac{π}{3}$ $pi / 3$ and $\frac{π}{2}$ $pi / 2$ . If one side of the triangle has a length of $9$ $9$ , what is the longest possible perimeter of the triangle?