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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