Question

In Figure 4, a circle is inscribed in a triangle PQR with PQ = 10 cm, QR = 8 cm and PR = 12 cm. Find the lengths QM, RN and PL.

Answer 1

We know that tangent segments to a circle from the same external point are congruent.
Now, we have
PL = PN, QL = QM and RM = RN
Now, PL + LQ = 10 cm .....(1)
PN + NR = 12 cm
⇒ PL + NR = 12 cm .....(2)
QM + MR = 8 cm
⇒ LQ + NR = 8 cm .....(3)
Adding all these we get
PL + LQ + PL + NR + LQ + NR = 30
⇒2(PL + LQ + NR) = 30
⇒PL + LQ + NR = 15 cm .....(4)
Solving (1) and (4), we get
NR = 5 cm
Solving (2) and (4), we get
LQ = 3 cm
Solving (3) and (4), we get
and PL = 7 cm
∴ PL = PN = 7 cm, QL = QM = 3 cm and RM = RN = 5 cm