From a point T, the length of the tangent to a circle is 32 cm and the distance of T from the centre is 40 cm. Find the diameter of the circle in cm.

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Accepted Answer

Let O be the center of the circle and T be the external point. OT = 40 cm (distance from center to point T), and the length of the tangent from T to the circle is 32 cm.

In a circle, the radius is perpendicular to the tangent at the point of contact. Therefore, triangle OTB (where B is the point of tangency) is a right triangle with OT as the hypotenuse, OB as the radius, and TB as the tangent length.

Using Pythagoras theorem:
OT^2 = OB^2 + TB^2
=> 40^2 = OB^2 + 32^2
=> 1600 = OB^2 + 1024
=> OB^2 = 1600 - 1024 = 576
=> OB = sqrt(576) = 24 cm

Diameter = 2 * radius = 2 * 24 = 48 cm.

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