A triangle has two corners with angles of $\frac{π}{3}$ $( pi ) / 3$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $17$ $17$ , what is the largest possible area of the triangle?

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Answer 1 · 2018-06-02T11:01:07

Longest possible Perimeter $P = 17 + 17 \sqrt{3} + 34 = 80.44$ $color(crimson)(P = 17 + 17sqrt3 + 34 = 80.44$

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

To get the longest perimeter, side 17 should correspond to the least angle $ˆ B = \frac{π}{6}$ $hat B = pi/6$

$\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}$ $a / sin A = b / sin B = c / sin C$ as per the Law of Sines.

$a = \frac{sin A \cdot b}{sin B} = \frac{sin (\frac{π}{3}) \cdot 17}{sin (\frac{π}{6})}$ $a = (sin A * b)/sin B = (sin (pi/3) * 17) / sin (pi/6)$

$a = 17 \sqrt{3}$ $a = 17sqrt3$

$c = \frac{sin (\frac{π}{2}) \cdot 17}{sin (\frac{π}{6})} = 34$ $c = (sin (pi/2) * 17) / sin (pi/6) = 34$

Perimeter $P = 17 + 17 \sqrt{3} + 34 = 80.44$ $color(crimson)(P = 17 + 17sqrt3 + 34 = 80.44$

A triangle has two corners with angles of ( pi ) / 3 π3 ( pi ) / 3 and ( pi )/ 6 π6 ( pi )/ 6 . If one side of the triangle has a length of 17 1717 , what is the largest possible area of the triangle?