In a triangle ABC, points P and Q are on AB and AC, respectively, such that AP = 4 cm, PB = 6 cm, AQ = 5 cm and QC = 7.5 cm. If PQ = 6 cm, then find BC (in cm).

15

In \(\Delta \)ABC and \(\Delta \)APQ,

\(\frac{AP}{AB}\) = \(\frac{4}{10}\) or \(\frac{2}{5}\)

\(\frac{AQ}{AC}\) = \(\frac{5}{12.5}\) or \(\frac{2}{5}\)

So, \(\frac{AB}{AP}\) = \(\frac{AC}{AQ}\)

So, \(\Delta \)ABC is similar to  \(\Delta \)APQ,

So, \(\frac{AB}{AP}\) = \(\frac{AC}{AQ}\)

So, \(\frac{PQ}{BC}\) = \(\frac{2}{5}\)

⇒ \(\frac{6}{BC}\) = \(\frac{2}{5}\)

⇒ BC = 6 x \(\frac{5}{2}\)

⇒ BC = 15

Therefore, BC is 15 cm.