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In triangle ABC, the angle bisector of angle BAC meets AC at D.  If angle BAC = 60, angle ABC = 60, and AD = 24, then find the area of triangle ABC.

 Feb 10, 2023
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Let's call the area of triangle ABC "A".

Since angle BAC is equal to 60 degrees and angle ABC is equal to 60 degrees, triangle ABC is an equilateral triangle.

Since angle bisector BD divides the side AC into segments of length AD and CD, we know that CD = AC - AD = AC - 24.

Let's call the length of AC "c". Then, we can use the Pythagorean Theorem to find the length of BD:

BD = sqrt(c^2 - 24^2) = sqrt(c^2 - 576)

Now, we can use the formula for the area of a triangle:

A = (1/2)bh

where b is the length of the base (AC in this case) and h is the height (BD in this case).

So,

A = (1/2)c * sqrt(c^2 - 576)

We can simplify this expression to get the final answer.

 Feb 10, 2023

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