Question

In ΔPQR, QM is the altitude to hypotenuse PR. If PM=8, RM=12, find PQ, QR and QM.

Answer 1

According to Geometric Mean Theorem.

If h denotes the altitude in a right triangle and p and q the segments on the hypotenuse then the theorem can be stated as:

h= \sqrt{pq} .

See attachment

Answer 2

Given, m∠Q = 90 , RM = 12 and MP = 8
∴ RP = (RM + MP) = 20
Let QM is x
from Pythagoras theorem in ΔQMP, we get
QP² = x² + 8² = x² + 64
from Pythagoras theorem in ΔQMR, we get
QR² = x² + 12² = x² + 144
from Pythagoras theorem in ΔPQR, we get
RP² = QP² + QR²
RP² = (x² + 64) + (x² + 144)
⇒ (20)² = 208 + 2x²
⇒ 400 = 2x² + 208
⇒2x² = 192
⇒ = x = 4√6
hence,  QM = 4√6

PQ² = x² + 64
⇒ PQ² = (4√6)² + 64 = 96 + 64 = 160
⇒ PQ = 4√10

QR² = x² + 144
⇒ QR² = (4√6)² + 144 = 96 + 144 = 240
⇒ QR = 4√15

hence, PQ = 4√10
QR = 4√15
and QM = 4√6