Problem:
Point D is on side CB of triangle ABC. If β CAD=β DAB=60β,AC=3 and AB=6, then the length of AD is
Answer Choices:
A. 2
B. 2.5
C. 3
D. 3.5
E. 4
Solution:
Let AD=y. Since AD bisects β BAC, we have CDDBβ=ACABβ=2; so we may set CD=x,DB=2x as in the figure. Applying the Law of Cosines to β³CAD and β³DAB, we have
x2=32+y2β3y(2x)2=62+y2β6y.β
Subtracting 4 times the first equation from the second yields 0=β3y2+6y=β3y(yβ2). Since yξ =0,y=2.
Alternate solution. Extend CA to E so that BEβ₯DA as in the new figure. Then β³ABE is equilateral: β BEA=β DAC by corresponding angles, β ABE=β BAD by alternate interior angles, and β EAB=180ββ120β. Since β³BECβΌβ³DAC, we have BEDAβ=CECAβ, or 6DAβ=93β.
So DA=2.
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