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# Geometry

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Help me, this is due soon

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 6, 2023

#1
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We can start by using the angle bisector theorem to find the length of BC.

Let E be the point where the angle bisector of angle ABC intersects AC. Then, by the angle bisector theorem, we have:

BD/DC = AB/AC

Since angle ABC = 60, we have AB/AC = sin(60) = sqrt(3)/2.

Also, we know that AD = 24.

Using these facts, we can solve for BD and DC:

BD/DC = (sqrt(3)/2)

BD/24-BD = (sqrt(3)/2)

Simplifying this equation, we get:

BD = 12sqrt(3)

DC = 24 - BD = 12(2-sqrt(3))

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

Area = (1/2) * base * height

We know that the base is BC and the height is the perpendicular distance from B to AC.

To find the height, we can use the fact that angle BDC is a right angle since BD is the angle bisector of angle ABC.

Thus, we have:

Area = (1/2) * BC * BH

where BH is the height of triangle ABC.

Using the Pythagorean theorem, we can find BH:

BH^2 = BD^2 - DH^2

where DH = DC - AD = 12(2-sqrt(3)) - 24 = -12sqrt(3)

Substituting in the values we found earlier, we get:

BH^2 = (12sqrt(3))^2 - (-12sqrt(3))^2

BH^2 = 432

BH = 12sqrt(3)

Finally, we can find the area of triangle ABC:

Area = (1/2) * BC * BH

Substituting in the values we found earlier, we get:

Area = (1/2) * (12sqrt(3)) * (12sqrt(3))

Area = 216

Therefore, the area of triangle ABC is 216 square units.

Mar 6, 2023