Two corners of a triangle have angles of $\frac{5 π}{8}$ $(5 pi )/ 8$ and $\frac{π}{3}$ $( pi ) / 3$ . If one side of the triangle has a length of $4$ $4$ , what is the longest possible perimeter of the triangle?

Question

1182 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-11-10T22:48:10

The longest possible perimeter is, $p = 58.8$ $p = 58.8$

Let $∠ C = \frac{5 π}{8}$ $angle C = (5pi)/8$
Let $∠ B = \frac{π}{3}$ $angle B = pi/3$

Then $∠ A = π - ∠ B - ∠ C$ $angle A = pi - angle B - angle C$

$∠ A = π - \frac{π}{3} - \frac{5 π}{8}$ $angle A = pi - pi/3 - (5pi)/8$

$∠ A = \frac{π}{24}$ $angle A = pi/24$

Associate the given side with the smallest angle, because that will lead to the longest perimeter:

Let side a = 4

Use the law of sines to compute the other two sides:

$\frac{b}{sin (∠ B)} = \frac{a}{sin (∠ A)} = \frac{c}{sin (∠ C)}$ $b/sin(angleB) = a/sin(angleA) = c/sin(angleC)$

$b = a \frac{sin (∠ B)}{sin (∠ A)} \approx 26.5$ $b = asin(angleB)/sin(angleA) ~~ 26.5$

$c = a \frac{sin (∠ C)}{sin (∠ A)} \approx 28.3$ $c = asin(angleC)/sin(angleA) ~~ 28.3$

$p = 4 + 26.5 + 28.3$ $p = 4 + 26.5 + 28.3$

The longest possible perimeter is, $p = 58.8$ $p = 58.8$

Two corners of a triangle have angles of 5π8 (5 pi )/ 8 and π3 ( pi ) / 3 . If one side of the triangle has a length of 4 4 , what is the longest possible perimeter of the triangle?