SEAMO 2027 - Paper F - SET 02 - Self-study Question 19

Point $X$ is outside triangle $ABC$. Points $P,Q,R$ lie on $AB,BC,AC$, respectively, and $XP\perp AB$, $XQ\perp BC$, $XR\perp AC$. Given $BP=1$, $BQ=2$, $CQ=3$, $CR=4$, $AR=5$, find $AP$.