From an external point P a tangent PT is drawn to a circle of radius 8 cm. If the distance OP from the centre is 17 cm, find the length of the tangent PT.

Step 1: Identify given information and properties

We are given the radius of the circle, $OT = 8$ cm, and the distance from the external point P to the center O, $OP = 17$ cm. A key property of circles is that the tangent at any point is perpendicular to the radius through the point of contact. Therefore, the angle $\angle OTP$ formed by the radius OT and the tangent PT is $90^\circ$. This means $\triangle OTP$ is a right-angled triangle.

Step 2: Apply the Pythagorean Theorem

Since $\triangle OTP$ is a right-angled triangle with the right angle at T, we can apply the Pythagorean theorem. The hypotenuse is OP, and the other two sides are OT and PT.

Step 3: Substitute values and solve for PT

Now, we substitute the given values into the Pythagorean theorem. We have $OP = 17$ cm and $OT = 8$ cm. We then solve the equation for $PT$ by isolating $PT^2$ and taking the square root.

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