# Two corners of a triangle have angles of #pi / 3 # and # pi / 6 #. If one side of the triangle has a length of #9 #, what is the longest possible perimeter of the triangle?

What we have is a 30-60-90 Triangle.

To get the longest possible perimeter, let's assume that the given length is for shortest side.

A 30-60-90 triangle has the following ratios:

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To find the longest possible perimeter of the triangle, we use the law of sines to find the length of the other two sides and then calculate the perimeter.

Given: Angle A = π/3 Angle B = π/6 Side c = 9

Using the law of sines: [ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} ]

We know that ( \sin(\pi/3) = \sqrt{3}/2 ) and ( \sin(\pi/6) = 1/2 ).

So, for angle A: [ \frac{a}{\sin(\pi/3)} = \frac{9}{\sin(C)} ] [ \frac{a}{\sqrt{3}/2} = \frac{9}{\sin(C)} ] [ a = \frac{9 \cdot \sqrt{3}}{2} \cdot \frac{2}{\sqrt{3}} = 9 ]

For angle B: [ \frac{b}{\sin(\pi/6)} = \frac{9}{\sin(C)} ] [ \frac{b}{1/2} = \frac{9}{\sin(C)} ] [ b = 18 \cdot \frac{1}{2} = 9 ]

So, both remaining sides have a length of 9.

Therefore, the longest possible perimeter of the triangle is: [ P = 9 + 9 + 9 = 27 ]

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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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