PQ and RS are common tangents to two circles intersecting at A and B. A and B, when produced on both sides, meet the tangents PQ and RS at X and Y, respectively. If \(AB = 3\) cm and \(XY = 5\) cm, then PQ is:
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For two circles intersecting at points A and B, the common tangents PQ and RS satisfy the relation \(PQ^2 = XY^2 - AB^2\).
Consider two circles that intersect at points A and B. Let PQ and RS be their common tangents. Let X and Y be the points where the lines passing through A and B intersect the tangents PQ and RS, respectively. Based on the properties of intersecting circles and their common tangents, the length of PQ is determined by one of the following equations: \[PQ = XY - AB = 5 - 3 = 2 \quad \text{or} \quad PQ = XY + AB = 5 + 3 = 8\]However, an analysis of the provided figure and the properties of tangents reveals the accurate relationship to be: \[PQ^2 = XY^2 - AB^2 = 5^2 - 3^2 = 25 - 9 = 16 \implies PQ = 4 \text{ cm}\]