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Question
Four alternative answers for the following question is given. Choose the correct alternative.
Length of a tangent segment drawn from a point which is at a distance 12.5 cm from the centre of a circle is 12 cm, find the diameter of the circle.
Options
25 cm
24 cm
7 cm
14 cm
Solution
Let O be the centre of the circle and AB be the tangent segment drawn from an external point A touching the circle at B.
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
∴ ∠ABO = 90º
In right ∆ABO,
\[{OA}^2 = {AB}^2 + {OB}^2 \]
\[ \Rightarrow OB = \sqrt{{OA}^2 - {AB}^2}\]
\[ \Rightarrow OB = \sqrt{\left( 12 . 5 \right)^2 - \left( 12 \right)^2}\]
\[ \Rightarrow OB = \sqrt{156 . 25 - 144}\]
\[ \Rightarrow OB = \sqrt{12 . 25} = 3 . 5 cm\]
Radius of the circle = OB = 3.5 cm
∴ Diameter of the circle = 2 × Radius of the circle = 2 × 3.5 = 7 cm
Hence, the correct answer is 7 cm .
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