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

Longest possible perimeter

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To find the longest possible perimeter of the triangle, we need to consider the two given angles and the given side length. Using the law of sines, we can find the lengths of the other two sides of the triangle.

Let's denote the angles of the triangle as A, B, and C, and the corresponding side lengths as a, b, and c.

Given: Angle A = (5π) / 8 Angle B = π / 3 Side b = 12

Using the law of sines:

a / sin(A) = b / sin(B)

We can find the length of side a:

a / sin((5π) / 8) = 12 / sin(π / 3)

After finding the length of side a, we can use the law of sines again to find the length of side c:

c / sin(C) = b / sin(B)

Given that the sum of angles in a triangle is π radians, we can find angle C:

Angle C = π - Angle A - Angle B

Now that we have all three side lengths, we can calculate the perimeter of the triangle:

Perimeter = a + b + c

By finding the values of side lengths a and c using the law of sines and calculating the perimeter using the formula, we can determine the longest possible perimeter of the triangle.

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The longest possible perimeter of the triangle is ( \frac{61\pi}{24} + 12 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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