PQ and RS are common external tangents to two intersecting circles, meeting the circles at points of contact. The line segment AB is the common chord of the two circles and, when produced on both sides, meets the tangents PQ and RS at points X and Y respectively. If AB = 3 cm and XY = 5 cm, then what is the length (in cm) of the common tangent segment PQ?

Introduction / Context:
This geometry question involves two intersecting circles, their common chord, and common tangents. The line through the common chord is extended to meet the tangents at two external points. Using properties of symmetry and power of a point, along with a convenient coordinate model, you can derive the length of the common tangent segment PQ purely from AB and XY.

Given Data / Assumptions:
• Two circles intersect at points A and B, forming a common chord AB.
• PQ and RS are common external tangents to the two circles.
• The line through A and B, when extended, meets PQ at X and RS at Y.
• Length AB = 3 cm and XY = 5 cm.
• We assume the configuration is symmetric, with the two circles arranged symmetrically with respect to the line through AB.

Concept / Approach:
A standard way to handle such symmetric configurations is to place the centres of the two circles symmetrically on the horizontal axis, with the common chord AB vertical. The common external tangents are then horizontal. In that coordinate set up, AB is vertical, XY is also vertical, and PQ is horizontal. One can relate half of AB and half of XY to the radii and the horizontal separation of the circle centres, and then read off PQ.

Step-by-Step Solution:
Step 1: Model the two circles with centres at (−d, 0) and (d, 0), and equal radius r. Step 2: The common chord AB lies along the vertical line x = 0, with A at (0, a) and B at (0, −a). Then AB = 2a = 3, so a = 1.5. Step 3: Because both circles pass through A and B, the relation for either centre is d2 + a2 = r2. Step 4: The external tangents become horizontal lines y = r and y = −r. The line AB extended meets these tangents at X = (0, r) and Y = (0, −r), so XY = 2r = 5, giving r = 2.5. Step 5: From d2 + a2 = r2, substitute a = 1.5 and r = 2.5: d2 = r2 − a2 = 2.52 − 1.52 = 6.25 − 2.25 = 4. Step 6: Hence d = 2. The tangent line y = r touches the left circle at (−d, r) and the right circle at (d, r). Step 7: Therefore the distance between these points, which is the length PQ, is 2d = 2 × 2 = 4 cm.

Verification / Alternative check:
Check that the model satisfies all given conditions: AB = 2a = 3 cm, XY = 2r = 5 cm, and both circles have radius 2.5 cm with centres at (−2, 0) and (2, 0). The tangent length between the two points of tangency is indeed 4 units horizontally. This confirms the correctness of PQ = 4 cm.

Why Other Options Are Wrong:
• 3 cm and 5 cm correspond to AB and XY respectively, but there is no reason for PQ to equal either of those directly.
• 2 cm would correspond to only the separation d, not twice d, so it misses the full tangent length connecting the two circles.

Common Pitfalls:
Visualizing the configuration can be challenging, and it is easy to confuse which segment corresponds to which length. Another common mistake is to try to use power of a point formulas directly without recognizing the convenient symmetry that allows a coordinate geometry interpretation.

Final Answer:
The length of the common tangent segment PQ is 4 cm.