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.
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Step-by-Step Solution
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 ∠OTP formed by the radius OT and the tangent PT is 90∘. This means △OTP is a right-angled triangle.
Step 2: Apply the Pythagorean Theorem
Since △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 PT2 and taking the square root.