Question : What is the length (in cm) of the transverse common tangent between two circles with radii 6 cm and 4 cm, given that the distance between their centres is 14 cm?
Option 1: $2 \sqrt{6}$
Option 2: $4 \sqrt{6}$
Option 3: $5 \sqrt{6}$
Option 4: $3 \sqrt{6}$
Correct Answer: $4 \sqrt{6}$
Solution :
The length of the transverse common tangent between two circles can be found using the formula:
$\text{Length} = \sqrt{{d^2 - (r_1 + r_2)^2}}$
Where: $d$ is the distance between the centres of the two circles,
\(r_1\) and \(r_2\) are the radii of the two circles.
Given that \(d = 14\) cm, \(r_1 = 6\) cm, and \(r_2 = 4\) cm, we can substitute these values into the formula:
$\text{Length} = \sqrt{{14^2 - (6 + 4)^2}} = \sqrt{{196 - 100}} = \sqrt{{96}} =\sqrt{{16\times6}} = 4\sqrt{{6}}\text{ cm}$
Hence, the correct answer is $4 \sqrt{6}$.
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