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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


Team Careers360 25th Jan, 2024
Answer (1)
Team Careers360 27th Jan, 2024

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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