Problem:
In △ABC we have AB=7,AC=8, and BC=9. Point D is on the circumscribed circle of the triangle so that AD bisects ∠BAC. What is the value of AD/CD?
Answer Choices:
A. 89​
B. 35​
C. 2
D. 717​
E. 25​
Solution:
Suppose that AD and BC intersect at E.
Since ∠ADC and ∠ABC cut the same arc of the circumscribed circle, the Inscribed Angle Theorem implies that
∠ABC=∠ADC.
Also, ∠EAB=∠CAD, so △ABE is similar to △ADC, and
CDAD​=BEAB​
By the Angle Bisector Theorem,
ECBE​=ACAB​,
so
BE=ACAB​⋅EC=ACAB​(BC−BE) and BE=AB+ACAB⋅BC​.
Hence
CDAD​=BEAB​=BCAB+AC​=97+8​=35​
Answer: B​.
The problems on this page are the property of the MAA's American Mathematics Competitions
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