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# geometry

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In triangle ABC, the angle bisector of BAC meets BC at D. If BAC=60 degrees, ABC=45 degrees, and AD=24, then find the area of ABC.

I think I'm supposed to build triangles, I don't know where and what triangles to create though. Sep 17, 2023

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Since BAC=60 degrees and ABC=45 degrees, then ACB=60 + 45 = 105 degrees.

Since the angle bisector of BAC meets BC at D, we know that BD/CD = AB/AC. We also know that AB = 24 + 15 = 39 and AC = 24. Substituting these values, we get:

BD/CD = AB/AC BD/CD = 39/24 BD = (39/24) * CD

We can use the Law of Sines to solve for CD:

sin(ABC) / AC = sin(ACB) / CD sin(45) / 24 = sin(105) / CD CD = (24 * sin(105)) / sin(45) CD = 30

Now that we know BD = (39/24) * CD = (39/24) * 30 = 52.5 and CD = 30, we can use the formula for the area of a triangle to find the area of ABC:

Area of ABC = (1/2) * BD * CD Area of ABC = (1/2) * 52.5 * 30

Area of ABC = 787.5

Therefore, the area of triangle ABC is 787.5 square units.

Sep 17, 2023