# An isosceles triangle has sides A, B, and C, such that sides A and B have the same length. Side C has a length of #16 # and the triangle has an area of #108 #. What are the lengths of sides A and B?

Denoting the height as

Using the Pythagorean Theorem with non-hypotenuse sides

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Using the formula for the area of a triangle ( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} ), and knowing that the triangle is isosceles, we can set up the equation:

[108 = \frac{1}{2} \times A \times h ]

where ( A ) and ( B ) represent the equal sides of the triangle, and ( C ) is the base.

We can rearrange this equation to solve for the height ( h ):

[h = \frac{2 \times 108}{C} = \frac{2 \times 108}{16} = 13.5]

Now, using the Pythagorean theorem for one of the two right triangles formed by dropping an altitude from the apex to the base:

[A^2 = h^2 + (\frac{C}{2})^2] [A^2 = 13.5^2 + 8^2] [A^2 = 182.25 + 64] [A^2 = 246.25] [A = \sqrt{246.25}] [A = 15.7]

So, sides (A) and (B) have lengths of approximately (15.7) units.

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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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