Question : A transverse common tangent is drawn to two circles of radius 8.5 cm and 5.5 cm. The centres of the two circles are 18 cm apart. What is the length (in cm) of the tangent?
Option 1: $10 \sqrt{3}$
Option 2: $10 \sqrt{2}$
Option 3: $8 \sqrt{2}$
Option 4: $8 \sqrt{3}$
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Correct Answer: $8 \sqrt{2}$
Solution :
Given: The Radii of circles are 8.5 cm and 5.5 cm and the length between circles is 18 cm.
$r_1=8.5\ \text{cm}$
$r_2=5.5\ \text{cm}$
$d=18\ \text{cm}$
The length of a transverse common tangent
$=\sqrt{d^2−(r_1+r_2)^2}$
$=\sqrt{(18)^2−(8.5+5.5)^2}$
$=\sqrt{324−196}$
$=\sqrt{128}=8 \sqrt{2}\ \text{cm}$
Hence, the correct answer is $8 \sqrt{2}\ \text{cm}$.
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