Question

Prove that : \(PM = \frac{1}{2} (PQ + QR + PR)\)

Hint:

For any triangle, the area can also be expressed as \(Area = r \times s\), where \(r\) is inradius and \(s\) is semi-perimeter. This is another great way to find the radius!

Answer 1

Step 1: Solving OR (B):
1. We know tangents from external points are equal: \(PM = PN\), \(QM = QS\), and \(RN = RS\). 2. Perimeter of \(\triangle PQR = PQ + QR + PR = PQ + (QS + SR) + PR\). 3. Substitute tangents: Perimeter \(= PQ + QM + RN + PR = (PQ + QM) + (PR + RN)\). 4. Since \(PQ + QM = PM\) and \(PR + RN = PN\), Perimeter \(= PM + PN = 2PM\). 5. Therefore, \(PM = \frac{1}{2} (\text{Perimeter})\).
Step 2: Final Answer (OR):
\(PM = \frac{1}{2} (PQ + QR + PR)\) is proved.