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

 Feb 26, 2024

If angle BAC = angle ABC = 60 degrees, then angle BCA = 60 degrees by triangle interior angle sum, and thus triangle ABC is equilateral. The angle bisector of the apex of an isosceles triangle (which includes the equilateral triangle) is also the height of the triangle. Since AD = 24, and angle A is being bisected, AD is also the height with length 24. By 30 - 60 - 90, the base of triangle ABC has length BC = 8sqrt(3), and the area is b*h/2 = 96sqrt(3) units^2.

 Feb 26, 2024

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