Question : Two circles with their centres at O and P and radii 8 cm and 4 cm respectively touch each other externally. The length of their common tangent is:
Option 1: 8.5 cm
Option 2: $\frac{8}{\sqrt{2}}$ cm
Option 3: $8\sqrt{2}$ cm
Option 4: 8 cm
Correct Answer: $8\sqrt{2}$ cm
Solution : Given, OQ = 8 cm and PR = 4 cm Construction: Draw a perpendicular from P to OQ. Since QR is a tangent, $\angle$ OQR = $\angle$ PRQ = 90$^\circ$ So, PSQR forms a rectangle. Applying Pythagoras theorem in $\triangle$ POS, PO 2 = OS 2 + PS 2 ⇒ (8+4) 2 = (8-4) 2 + QR 2 ⇒ QR 2 = 144 – 16 ⇒ QR = $8\sqrt{2}$ cm. Hence, the correct answer is $8\sqrt{2}$ cm.
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