Question

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

Answer 1

Let's write the ratio 2:5 as 2x:5x and 2:3 as 2y:3y.

ABCD is a parallelogram, so AB=CD. That means that 2x+5x=2y+3y.

From this equation we obtain that y=1.4x.

Now we have 2.8x:4.2x instead of 2y:3y.

Let's take a look at triangles APR and CQR(look at the drawing below).

Angles PRA and QRC are equal. So are angles RAP and RCQ.

That means that triangles APR and CQR are similar(according to the first theorem of the similar triangles).

That means that the corresponding sides of triangles keep the same ratio:

AP:CQ = 2:2.8

AR:CR=2:2.8

PR:QR=2:2.8