Two corners of a triangle have angles of $\frac{π}{3}$ $pi / 3$ and $\frac{π}{6}$ $pi / 6$ . 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 · 2016-12-04T07:55:28

$P = 27 + 9 \sqrt{3}$ $P = 27 + 9sqrt3$

What we have is a 30-60-90 Triangle.

To get the longest possible perimeter, let's assume that the given length is for shortest side.

A 30-60-90 triangle has the following ratios:

$30 : 60 : 90 = x : \sqrt{3} x : 2 x$ $30:60:90 = x: sqrt3x : 2x$

$x = 9$ $x = 9$
$\Rightarrow \sqrt{3} x = 9 \sqrt{3}$ $=> sqrt3x = 9sqrt3$
$\Rightarrow 2 x = 18$ $=> 2x = 18$

$P = S_{1} + S_{2} + S_{3}$ $P = S_1 + S_2 + S_3$

$P = 9 + 9 \sqrt{3} + 18$ $P = 9 + 9sqrt3 + 18$

$P = 27 + 9 \sqrt{3}$ $P = 27 + 9sqrt3$

Two corners of a triangle have angles of pi / 3 π3pi / 3 and pi / 6 π6 pi / 6 . If one side of the triangle has a length of 9 99 , what is the longest possible perimeter of the triangle?