A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Find the length of PQ.

Step 1: Identify the given information

We are given the radius of the circle, which is the distance from the center O to the point of tangency P. We are also given the distance from the center O to the external point Q.

Step 2: Recall the Tangent-Radius Theorem

According to the Tangent-Radius Theorem, the tangent at any point of a circle is perpendicular to the radius through the point of contact. Therefore, the radius OP is perpendicular to the tangent PQ at point P.

Step 3: Form a right-angled triangle

Since OP is perpendicular to PQ, the triangle formed by O, P, and Q (triangle OPQ) is a right-angled triangle with the right angle at P.

Step 4: Apply the Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Here, OQ is the hypotenuse.

Step 5: Substitute the given values

Substitute the known values of OQ = 12 cm and OP = 5 cm into the Pythagorean theorem equation.

Step 6: Calculate the squares

Perform the squaring operations: $12^2 = 144$ and $5^2 = 25$.

Step 7: Isolate PQ squared

Subtract 25 from both sides of the equation to isolate the term involving $PQ^2$.

Step 8: Calculate PQ squared

Perform the subtraction: $144 - 25 = 119$.

Step 9: Find PQ

Take the square root of both sides to find the length of PQ. Since PQ represents a length, we consider only the positive square root.

Step 10: State the unit

The length of PQ is $\sqrt{119}$ centimeters.

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