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: Using Property of Tangents:
From an external point, the lengths of tangents drawn to a circle are equal.

Therefore:
PM = PN
QM = QS
RN = RS

Step 2: Writing the Perimeter of ΔPQR:
Perimeter of ΔPQR = PQ + QR + PR

Now break side QR into two parts:
QR = QS + SR

So,
Perimeter = PQ + (QS + SR) + PR

Step 3: Substituting Equal Tangents:
Since QM = QS and RN = SR,

Perimeter = PQ + QM + RN + PR

Group the terms:
= (PQ + QM) + (PR + RN)

Step 4: Using Tangent Equalities Again:
From property of tangents:
PQ + QM = PM
PR + RN = PN

So perimeter becomes:
Perimeter = PM + PN

But PM = PN,

Perimeter = PM + PM
= 2PM

Step 5: Final Result:
2PM = Perimeter of ΔPQR

Therefore,
PM = 1/2 (PQ + QR + PR)

Final Conclusion:
Hence, it is proved that the tangent length PM is half of the perimeter of ΔPQR.