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Question:

In ΔPQR, m∠Q = 90° and QM is an altitude; M ∈ PR. If QM = 12, PR = 26; then find PM and RM; If PM < RM; then find PQ and QR.

Solution:

In ΔPQR, m∠Q = 90° and QM is an altitude; M ∈ PR ∴ PM⋅MR = QM²
Suppose PM = a ∴ RM = PR - PM = 26 - a
Now, QM² = PM⋅RM ⇒ 12² = a(26 - a) ⇒ 144 = 26a - a² ⇒ a² - 26a + 144 = 0 ⇒ (a - 8)(a - 18) = 0 ⇒ a = 8 or a = 18
So PM = 8 and RM = 18 or PM = 18 and RM = 8
If PM < RM, then PM = 8 and RM = 18
PQ² = PM⋅PR = 8 × 26 = 16 × 13 ⇒ PQ = 4√13
QR² = RM⋅PR = 18 × 26 = 36 × 13 ⇒ QR = 6√13