The δpqr is right-angled at p, and pn is an altitude. if qn = 12 in and nr = 6 in, find pn, pq, pr.

In the right triangle PQR:
Hypotenuse: QR = QN + NR = 12 + 6 = 18 cm
PN² = QN · NR
PN² = 12 · 6 = 72
PN = √72 = √(36 · 2 ) = 6√2 cm
PQ² = (6√2)² + 12² = 72 + 144 = 216
PQ = √216 = √(36 · 6 ) = 6√6 cm
PR² = 18² - (6√6)² = 324 - 216 = 108
PR = √108 = √(36 · 3) = 6√3 cm

PN = 6√2 cm,
PQ = 6√6 cm,
PR = 6√3 cm.