In the following figure, O is the centre of the circle. If OP = 5 cm and radius of the circle is 3 cm, find the length of the tangent PT.

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  1. cm
  2. 4 cm
  3. cm
  4. cm

Answer (Detailed Solution Below)

Option 2 : 4 cm
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Detailed Solution

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Given:

O is the center of the circle.

OP = 5 cm

Radius of the circle (OT) = 3 cm

PT is a tangent to the circle at point T.

Formula Used:

The radius to the point of tangency is perpendicular to the tangent. Therefore, ∠OTP = 90°.

Pythagorean theorem: In a right-angled triangle, (hypotenuse)2 = (base)2 + (perpendicular)2

In right-angled triangle OTP, OP is the hypotenuse, OT and PT are the other two sides.

Calculation:

In right-angled triangle OTP, by the Pythagorean theorem:

OP2 = OT2 + PT2

52 = 32 + PT2

25 = 9 + PT2

PT2 = 25 - 9

PT2 = 16

PT = \(\sqrt{16}\) = ±4 

Since length cannot be negative, PT = 4 cm.

The length of the tangent PT is 4 cm.

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