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

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

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Answer 1 · 2016-10-22T21:17:21

The perimeter is 61

Explanation:

Let $/_A = pi/12$
Let $/_B = (5pi)/8$
Then $/_C = pi - (5pi)/8 - pi/12 = (24pi)/24 - (15pi)/24 - (2pi)/24 = (7pi)/24$

Let side $a$ be the side opposite $/_A$ with length $8$ . We do this because associating the given side with the smallest angle will yeild the largest perimeter.

Using the Law of Sines we can write the equation for sides $b$ and $c$ :

$b = asin(/_B)/sin(/_A)$
$c = asin(/_C)/sin(/_A)$

Let p = the perimeter

$p = a + a(sin(/_B) + sin(/_C))/sin(/_A)$

$p = 61$

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