Question

In the right-angled triangle QPR, PM is an altitude.

Given that QR = 8 cm and MQ = 3.5 cm, calculate the value of PR.

Answer 1

In ΔPQM and ΔQPR,

∠PMQ = ∠QPR   ...(Each = 90°)

∠Q = ∠Q   ...(Common)

∴ ΔPQM ∼ ΔQPR  ...(AA postulate)

∴ `(PQ)/(QR) = (QM)/(PQ) = (PM)/(PR)`  ...(i)

`\implies` PQ² = QR × QM

= 8 × 3.5

= 28

∴ `PQ = sqrt(28)`  ...(ii)

In ΔPQR, ∠P = 90° and PM ⊥ QR

∴ PM² = QM × MR = 3.5 × 4.5   ...(∴ MR = QR – QM)

∴ `PM = sqrt(3.5 xx 4.5)`   ...(iii)

From (i) `(PQ)/(QR) = (PM)/(PR)`

`sqrt(28)/8 = sqrt(3.5 xx 4.5)/(PR)`

Squaring on both sides,

`28/64 = (3.5 xx 4.5)/(PR)`

`PR^2 = (3.5 xx 4.5 xx 64)/28`

= `(35 xx 45 xx 64)/(10 xx 10 xx 28)`

= `10080/2800`

`=>` PR² = 36 = (6)²

∴ PR = 6 cm.