Two corners of a triangle have angles of $\frac{5 π}{8}$ $(5 pi )/ 8$ and $\frac{π}{12}$ $( pi ) / 12$ . If one side of the triangle has a length of $3$ $3$ , what is the longest possible perimeter of the triangle?

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Two corners of a triangle have angles of $\frac{5 π}{8}$ $(5 pi )/ 8$ and $\frac{π}{12}$ $( pi ) / 12$ . If one side of the triangle has a length of $3$ $3$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2016-10-15T00:32:33

The maximum perimeter is 22.9

Explanation:

The maximum perimeter is achieved, when you associate the given side with the smallest angle.

Calculate the third angle:
$\frac{24 π}{24} - \frac{15 π}{24} - \frac{2 π}{24} = \frac{7 π}{24}$ $(24pi)/24 - (15pi)/24 - (2pi)/24 = (7pi)/24$

$\frac{π}{12}$ $pi/12$ is the smallest

Let angle $A = \frac{π}{12}$ $A = pi/12$ and the length of side $a = 3$ $a = 3$
Let angle $B = \frac{7 π}{24}$ $B = (7pi)/24$ . The length of side b is unknown
Let angle $C = \frac{5 π}{8}$ $C = (5pi)/8$ . The length of side c is unknown.

Using the law of sines:

The length of side b:

$b = 3 \frac{sin (\frac{7 π}{24})}{sin (\frac{π}{12})} \approx 9.2$ $b = 3sin((7pi)/24)/sin(pi/12) ~~ 9.2$

The length of side c:

$c = 3 \frac{sin (\frac{5 π}{8})}{sin (\frac{π}{12})} \approx 10.7$ $c = 3sin((5pi)/8)/sin(pi/12) ~~ 10.7$

P = 3 + 9.2 + 10.7 = 22.9

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Two corners of a triangle have angles of $\frac{5 π}{8}$ $(5 pi )/ 8$ and $\frac{π}{12}$ $( pi ) / 12$ . If one side of the triangle has a length of $3$ $3$ , what is the longest possible perimeter of the triangle?

1 Answer

Explanation:

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Impact of this question

Science

Math

Humanities

... and beyond

Two corners of a triangle have angles of 5π8 (5 pi )/ 8 and π12 ( pi ) / 12 . If one side of the triangle has a length of 3 3 , what is the longest possible perimeter of the triangle?

1 Answer

Explanation:

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Impact of this question

Two corners of a triangle have angles of $\frac{5 π}{8}$ $(5 pi )/ 8$ and $\frac{π}{12}$ $( pi ) / 12$ . If one side of the triangle has a length of $3$ $3$ , what is the longest possible perimeter of the triangle?