Answer to Question #94687 in Geometry for Emmanuel Daniels

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

$AK||PQ$ and $PQ||LC$

$AO=h$ (height)

$APR$ and $CQR$ similar triangles $AR/RC=PR/RQ$

$AP=2*X\\ PB=5*X\\ DQ=3*Y\\ QC=2*Y\\ 2*X+5*X=3*Y+2*Y\\ X/Y=5/7\\ Area APQK=2*X*h\\ Area PLCQ=2*Y*h\\ AreaAPQK/AreaPLCQ=AR/RC=PR/RQ=(2*X*h)/(2*Y*h)=X/Y=5/7$

Answer: $AR/RC=PR/RQ=5/7$