(c) √5.
Join P′Q′.
P′, Q′ being the mid-points of QR and PR respectively, we have P′Q′ || PQ and P′Q′ = \(\frac{1}{2}\) PQ
(By the mid-point theoram)
Let OP′ = a, OQ′ = b, OP = c, OQ = d and PQ = x
Then, P′Q′ = \(\frac{1}{2}\) \(x\) .
∴ In rt. Δ OP′Q′,
a2 + b2 = \(\frac{2}{4}\) \(x\) …(i)
In rt. Δ OP′Q,
a2 + d2 = \(\frac{9}{4}\) …(ii)
In rt. Δ OQP,
c2 + d2 = x2 …(iii)
In rt. Δ OQ′P,
b2 + c2 = 4 …(iv)
∴ Eq (i) – Eq (ii) + Eq (iii) – Eq (iv)
⇒ a2 + b2 – (a2 + d2) + (c2 + d2) – (b2 + c2) = \(\frac{x^2}{4}\) - \(\frac{9}{4}\) + x2 - 4
⇒ 0 = \(\frac{5x^2}{4}-\frac{25}{4}\) ⇒ \(\frac{5x^2}{4}=\frac{25}{4}\) ⇒ x2 = 5 ⇒ x = √5.
