Question

A circle is inscribed in ΔABC, touching AB, BC and AC at P, Q and R, respectively. If AB = 10 cm, AR = 7 cm and CR = 5 cm, find the length of BC.

Answer 1

$$\begin{gathered} \text{Given, a circle inscribed in triangle ABC, such that the circle touches the sides of the triangle at P,Q,R}\text{.} \\ \text{Tangents drawn to a circle from an external point are equal}\text{.} \\ \therefore \text{AP}=\text{AR}=7\text{cm, CQ}=\text{CR}=5\text{cm}\text{.} \\ \mathrm{Now},\mathit{\text{BP}}=(\mathit{\text{AB}}-\mathit{\text{AP}})=(10-7)=3\text{cm} \\ \therefore \mathit{\text{BP}}=\mathit{\text{BQ}}=3\text{cm} \\ \therefore \mathit{\text{BC}}=(\mathit{\text{BQ}}+\mathit{\text{QC}}) \\ =>BC=3+5 \\ =>BC=8 \\ \therefore \text{The length of}\mathit{\text{BC}}\text{is 8 cm}\text{.} \end{gathered}$$