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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.

 Mar 4, 2023
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Since AD is the angle bisector of angle BAC, we have:

angle BAD = angle CAD = 30

Therefore, triangle ABD is a 30-60-90 triangle, which means that:

BD = AD / sqrt(3) = 8 * sqrt(3)

Now, since angle ABC = 60, triangle ABC is equilateral, which means that all sides are equal. Therefore, we have:

AB = BC

Let x be the length of AB (which is also the length of BC). Then, using the law of cosines in triangle ABC, we have:

x^2 = 24^2 + (8 * sqrt(3))^2 - 2(24)(8 * sqrt(3))(cos 60)
x^2 = 576 + 192 - 384
x^2 = 384
x = 8 * sqrt(6)

Therefore, the area of triangle ABC is:

(area of equilateral triangle with side length 8 * sqrt(6))
= (sqrt(3) / 4) * (8 * sqrt(6))^2
= 96 * sqrt(3)

Therefore, the area of triangle ABC is 96 * sqrt(3).

 Mar 5, 2023

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