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Challenge Problem 2 2016

Triangle 1

(Note: Not drawn to scale)

In triangle ABC, CE is perpendicular to AB and BD is perpendicular to to AC. If EC = 5 and BD = 3, what is the ratio of AC to AB?

(A) 3 : 5          (B) 3 : 4          (C) 4 : 5          (D) 1 : 1          (E) 5 : 3

Every triangle has three altitudes and the area of the triangle is always the same regardless of the altitude we use to calculate it. This is also true of the base. In this case AB and AC are both bases. CE is the altitude corresponding the AB, and BD is the altitude corresponding to base AC. Consequently, we can calculate the area in two ways:

\(A = \)\(\Large\frac{1}{2}\)\(AB\times{CE} = \)\(\Large\frac{1}{2}\)\(AC\times{BD}\)

Cancel the fractions and we get

\(AB\times{CE} = AC\times{BD}\)

Insert our values for AC and BD and we get

\(5AC = 3BD\)

So this ratio of of AC to BD is 5 to 3. The correct answer is (E)