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.
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Step-by-Step Solution
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: 122=144 and 52=25.
Step 7: Isolate PQ squared
Subtract 25 from both sides of the equation to isolate the term involving PQ2.
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 119 centimeters.