A triangle has two corners with angles of $\frac{π}{2}$ $pi / 2$ and $\frac{π}{8}$ $( pi )/ 8$ . If one side of the triangle has a length of $13$ $13$ , what is the largest possible area of the triangle?

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A triangle has two corners with angles of $\frac{π}{2}$ $pi / 2$ and $\frac{π}{8}$ $( pi )/ 8$ . If one side of the triangle has a length of $13$ $13$ , what is the largest possible area of the triangle?

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Answer 1 · 2018-04-16T00:26:07

The area is $\frac{169}{2 (\sqrt{2} - 1)} \approx 204$ $169/(2(sqrt(2)-1))~~204$

Explanation:

Original Drawing

If one of the angles of the triangle is $\frac{π}{2}$ $pi/2$ , it is a right triangle. The measure of the third angle of the triangle must be $\frac{π}{2} - \frac{π}{8} = \frac{3 π}{8}$ $pi/2-pi/8=(3pi)/8$ . The shortest side of this right triangle will be the leg that is opposite the angle measuring $\frac{π}{8}$ $pi/8$ . Let $x$ $x$ = the length of the longer leg. The area, $A$ $A$ , of the triangle will be

$A = \frac{13}{2} x$ $A=13/2x$

From trigonmetry

$x = \frac{13}{tan (\frac{π}{8})}$ $x=13/tan(pi/8)$

So

$A = \frac{13^{2}}{2 tan (\frac{π}{8})} = \frac{169}{2 (\sqrt{2} - 1)} \approx 204$ $A=13^2/(2tan(pi/8))=169/(2(sqrt(2)-1))~~204$

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A triangle has two corners with angles of $\frac{π}{2}$ $pi / 2$ and $\frac{π}{8}$ $( pi )/ 8$ . If one side of the triangle has a length of $13$ $13$ , what is the largest possible area of the triangle?

1 Answer

Explanation:

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A triangle has two corners with angles of $\frac{π}{2}$ $pi / 2$ and $\frac{π}{8}$ $( pi )/ 8$ . If one side of the triangle has a length of $13$ $13$ , what is the largest possible area of the triangle?