Answer to Question #94909 in Geometry for Mashud

Question

In a parallogram ABCD, P divides AB in the ratio 2 : 5 and Q divides DC in
the ratio 3 : 2. If AC and P Q intersect at R, find the ratios AR : RC and P R : RQ.

Answer 1

1) ABCD is a parallogram, it means that AB || CD, BC || AD.

2) P divides AB in the ratio 2 : 5. Then let AP equals 2x, and PB equals 5x, where x is a specific unit of measure.

The same with DC. Q divides DC in the ratio 3 : 2. Then let DQ equls 3y, QC equls 2y, where y is another specific unit of measure.

3)PQ intersects AC in R, that's why the angle PRA and the angle CRQ are vertical, that's why they are of equal degrees.

4) From (1): AB || CD, AC is secant, then angles RCQ and PAR are equal and the corresponding triangles are similar.

It means that $\frac{AP}{CQ} = \frac{PR}{RQ} = \frac{AR}{RC} = \frac{x}{y}$

We know that ABCD is a parallogram, it means that AB = CD $\implies$ 2x + 5x = 2y + 3y $\implies$ 7x = 5y

Then

ANSWER: $\frac{PR}{RQ} = \frac{AR}{RC} = \frac{x}{y} = \frac{5}{7}$