Exam

Olympiad workbook

Subject

IOQM

Chapter

Circle

Olympiad workbook · IOQM · Circle

Let A B C D be a trapezium in which A B \| C D and A D A B ....

Let $A B C D$ be a trapezium in which $A B \| C D$ and $A D \perp A B$. Suppose $A B C D$ has an incircle which touches AB at Q and CD at P . Given that $\mathrm{PC}=36$ and $\mathrm{QB}=49$, Find PQ.

Step-by-step solution

Let incircle touch $B C$ at $R$ So, $\mathrm{CR}=36, \mathrm{BR}=49$ Further let inradius $=r$ So, $\mathrm{AQ}=\mathrm{PD}=\mathrm{r}$ \& $\mathrm{AD}=2 \mathrm{r}$ Let perpendicular from C meet AD at S So, $B S=13, B C=85$ Now, $(\mathrm{CS})^2=85^2-13^2=98 \times 72=49 \times 144$ So $\mathrm{CS}=7 \times 12=84$ Hence $\mathrm{PQ}=84$