SOLUTION: In triangle ABC ,PQ is parallel to AC and PQ divides triangular region ABC into two parts such that ar(BPQ)=1/4 at(PQCA). Find BP:PA.

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Question 1105846: In triangle ABC ,PQ is parallel to AC and PQ divides triangular region ABC into two parts such that ar(BPQ)=1/4 at(PQCA). Find BP:PA.
Answer by greenestamps(13200) About Me  (Show Source):
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If the area of triangle BPQ is 1/4 the area of trapezoid PQCA, then the area of triangle BPQ is 1/5 the area of triangle ABC.

Since triangles ABC and PBQ are similar, the ratio of their areas is the square of the scale factor between the two triangles.

Since the ratio of areas is 1:5, the scale factor is 1:sqrt(5); that makes BP:AB = 1:sqrt(5). And that makes BP:PA = 1:(sqrt(5)-1).