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 $1$ $1$ , 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 $1$ $1$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2016-11-23T20:02:25

The longest possible perimeter is $P \approx 10.5$ $P ~~ 10.5$

Explanation:

Let $∠ A = \frac{π}{12}$ $angle A = pi/12$
Let $∠ B = \frac{5 π}{8}$ $angle B = (5pi)/8$
Then $∠ C = π - \frac{5 π}{8} - \frac{π}{12}$ $angle C = pi - (5pi)/8 - pi/12$
$∠ C = \frac{7 π}{24}$ $angle C = (7pi)/24$

The longest perimeter occurs, when the given side is opposite the smallest angle:

Let side $a = the side opposite angle A = 1$ $a = "the side opposite angle A" = 1$

The perimeter is: $P = a + b + c$ $P = a + b + c$

Use the Law of Sines

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

to substitute into the perimeter equation:

$P = a \frac{1 + sin (B) + sin (C)}{sin (A)}$ $P = a(1 + sin(B) + sin(C))/sin(A)$

$P = 1 \frac{1 + sin (\frac{5 π}{8}) + sin (\frac{7 π}{24})}{sin (\frac{π}{12})}$ $P = 1(1 + sin((5pi)/8) + sin((7pi)/24))/sin(pi/12)$

$P \approx 10.5$ $P ~~ 10.5$

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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 $1$ $1$ , 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 1 1 , 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 $1$ $1$ , what is the longest possible perimeter of the triangle?