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4. In equilatral triangle ABC with side 3, D and E are points on AB and AC, respectively, such that AD = AE = 2. CD and BE intersect at F. What is the area of FBC?
AD = ED = 2 means AD is 4.
It is an equilateral triangle, so all of the other sides are 4.
Also, all of the angle are 60°.
Look at triangle AEF. It is a 30°-60°-90°.
If G is in the midpoint of BC,
then since we have an equilateral triangle, we know the EF=DF=GF.
We also know that CG=2 because of this.
Since this side of the 30°-60°-90° is known to be
√3 times the short side, and is also known to have length 2,
that means the short side is 2/√3 = 2√3/9.
Triangles CGF and BGF are the same and the area of each is
base*height/2. When these two areas are added together,
2/2 cancels and the total area is base*height.
The base is known to be 2 and the height is known to be 2√3/9,
so the total area is known to be 4√3/9.
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Scott A Wilson
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