Question : $\triangle$ PQR circumscribes a circle with centre O and radius r cm such that $\angle$ PQR = $90^{\circ}$. If PQ = 3 cm, QR = 4 cm, then the value of r is:
Option 1: 2
Option 2: 1.5
Option 3: 2.5
Option 4: 1
Correct Answer: 1
Solution :
PQ = 3 cm, QR = 4 cm
OB = OA = OC = r
Since tangents drawn from a point to the circle are equal,
AQ = QB = $r$
⇒ PA = PC = $3-r$
⇒ RC = RB = $4-r$
By Pythagoras theorem,
PR
2
= QR
2
+ PQ
2
⇒ PR = $\sqrt{9+16}$ = 5 cm
⇒ PR = PC + CR
⇒ 5 = $3-r$ + $4-r$
⇒ $2r$ = 7–5 = 2
⇒ $r$ = 1 cm
Hence, the correct answer is 1.
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