Question 1105846
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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.<br>
Since triangles ABC and PBQ are similar, the ratio of their areas is the square of the scale factor between the two triangles.<br>
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).