Two corners of a triangle have angles of $\frac{π}{8}$ $pi / 8$ and $\frac{π}{3}$ $pi / 3$ . If one side of the triangle has a length of $2$ $2$ , what is the longest possible perimeter of the triangle?

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Two corners of a triangle have angles of $\frac{π}{8}$ $pi / 8$ and $\frac{π}{3}$ $pi / 3$ . If one side of the triangle has a length of $2$ $2$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2016-04-06T15:32:25

The maximum perimeter is: $11.708$ $11.708$ to 3 decimal places

Explanation:

When ever possible draw a diagram. It helps to clarify what you are dealing with.

Tony B

Notice that I have labeled the vertices as with capital letters and the sides with small letter version of that for the opposite angle.

If we set the value of 2 to the smallest length then the sum of sides will be the maximum.

Using the Sine Rule

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

$\Rightarrow \frac{a}{sin (\frac{π}{8})} = \frac{b}{sin (\frac{13}{24} π)} = \frac{c}{sin (\frac{π}{3})}$ $=> a/(sin(pi/8))=b/(sin(13/24 pi)) = c/(sin(pi/3))$

Ranking these with the smallest sine value on the left

$\Rightarrow \frac{a}{sin (\frac{π}{8})} = \frac{c}{sin (\frac{π}{3})} = \frac{b}{sin (\frac{13}{24} π)}$ $=> a/(sin(pi/8))=c/(sin(pi/3)) = b/(sin(13/24 pi))$

So side $a$ $a$ is the shortest.

Set $a = 2$ $a=2$

$\Rightarrow c = \frac{2 sin (\frac{π}{3})}{sin (\frac{π}{8})} = 4.526$ $=> c=(2sin(pi/3))/(sin(pi/8))" "=" "4.526$ to 3 decimal places

$\Rightarrow b = \frac{2 sin (\frac{13}{24} π)}{sin (\frac{π}{8})} = 5.182$ $=> b=(2sin(13/24 pi))/(sin(pi/8))=5.182$ to 3 decimal places

So the maximum perimeter is: $11.708$ $11.708$ to 3 decimal places

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Two corners of a triangle have angles of $\frac{π}{8}$ $pi / 8$ and $\frac{π}{3}$ $pi / 3$ . If one side of the triangle has a length of $2$ $2$ , what is the longest possible perimeter of the triangle?

1 Answer

Explanation:

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Two corners of a triangle have angles of $\frac{π}{8}$ $pi / 8$ and $\frac{π}{3}$ $pi / 3$ . If one side of the triangle has a length of $2$ $2$ , what is the longest possible perimeter of the triangle?