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# Geometry

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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 18, 2023

Because angles bac and abc are 60 degrees, the triangle is equilateral. That also means that the height is AD (or 24). So now we just need to find the base, we can call the base x. Because AD is the angle bisect or of BAC, angles BDC is 90 degrees, so we can use the Pythagorean theorem on the half triangle. Using Pythagorean we get (x/2)^2 + 576 = x^2, or x^2 / 4 + 576 = x^ 2. That means 576 is 3x^2 / 4, or 3x^2 = 2304, so x equals sqrt768, but then we have to multiply the height and divide by two, so 24 x sqrt768 / 2, or 24sqrt768 / 2. That gives us our answer of 12$$\sqrt{728}$$