Problem:
In trapezoid ABCD,AB and CD are perpendicular to AD, with AB+CD= BC,AB<CD, and AD=7. What is AB⋅CD?
Answer Choices:
A. 12
B. 12.25
C. 12.5
D. 12.75
E. 13
Solution:
Let E be the foot of the perpendicular from B to CD. Then AB=DE and BE=AD=7. By the Pythagorean Theorem,
AD2=BE2=BC2−CE2=(CD+AB)2−(CD−AB)2=(CD+AB+CD−AB)(CD+AB−CD+AB)=4⋅CD⋅AB.
Hence, AB⋅CD=AD2/4=72/4=49/4=12.25.
Answer: B.
The problems on this page are the property of the MAA's American Mathematics Competitions
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