Two corners of a triangle have angles of $\frac{π}{2}$ $( pi )/ 2$ and $\frac{π}{6}$ $( pi ) / 6$ . If one side of the triangle has a length of $14$ $14$ , what is the longest possible perimeter of the triangle?

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Two corners of a triangle have angles of $\frac{π}{2}$ $( pi )/ 2$ and $\frac{π}{6}$ $( pi ) / 6$ . If one side of the triangle has a length of $14$ $14$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-05-12T04:41:05

$Longest Possible Perimeter = 14 + 24.25 + 28 = 66.25 units$ $color(green)("Longest Possible Perimeter" = 14 + 24.25 + 28 = 66.25 " units"$

Explanation:

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

To get the longest perimeter, side 14 should correspond to the least angle $\frac{π}{6}$ $pi/6$

Applying Law of Sines,

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

$\frac{14}{sin (\frac{π}{6})} = \frac{c}{sin (\frac{π}{3})}$ $14 / sin (pi/6) = c / sin (pi/3)$

$c = \frac{14 \cdot sin (\frac{π}{3})}{sin (\frac{π}{6})} = 24.25$ $c = (14 * sin (pi/3)) / sin (pi/6) = 24.25$

$a = \frac{14 \cdot sin (\frac{π}{2})}{sin (\frac{π}{6})} = 28$ $a = (14 * sin(pi/2)) / sin (pi/6) = 28$

$Perimeter P = a = b + c$ $color(green)(" Perimeter " P = a = b + c$

$Longest Possible Perimeter = 14 + 24.25 + 28 = 66.25 units$ $color(green)("Longest Possible Perimeter" = 14 + 24.25 + 28 = 66.25 " units"$

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Two corners of a triangle have angles of $\frac{π}{2}$ $( pi )/ 2$ and $\frac{π}{6}$ $( pi ) / 6$ . If one side of the triangle has a length of $14$ $14$ , 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 $\frac{π}{2}$ $( pi )/ 2$ and $\frac{π}{6}$ $( pi ) / 6$ . If one side of the triangle has a length of $14$ $14$ , what is the longest possible perimeter of the triangle?