A line PQ is drawn parallel to the side BC of Δ ABC which cuts side AB at P and side AC at Q. If AB = 9.0 cm, CA = 6.0 cm and AQ = 4.2 cm, find the length of AP.

In \[\vartriangle \text{ }APQ\text{ }and\vartriangle \text{ }ABC\]

\[\angle APQ\text{ }=\angle ABC\]

[As PQ || BC, corresponding angles are equal.]

\[\angle PAQ\text{ }=\angle BAC\]

[Common angle]

Hence, \[\vartriangle APQ\text{ }\sim\text{ }\vartriangle ABC\text{ }by\text{ }AA\] criterion for similarity

So, we have

\[AP/AB\text{ }=\text{ }AQ/AC\]

\[AP/9\text{ }=\text{ }4.2/6\]

Hence,

\[AP\text{ }=\text{ }6.3\text{ }cm\]