Two circles have centers O and O2 with radii 7 cm and 9 cm respectively and the distance between the centers is 20 cm; PQ is a transverse common tangent to the circles that cuts the line segment OO2 at X; find the length of segment O2X in centimetres.

Introduction / Context:
This geometry question involves two circles, a transverse common tangent, and the line segment joining the centers. It tests understanding of circle geometry, similar triangles, and relationships between radii, center distance, and tangents. The goal is to find the length of a segment on the line joining the centers where the tangent intersects that line.

Given Data / Assumptions:
- Circle 1 has center O and radius 7 cm.
- Circle 2 has center O2 and radius 9 cm.
- The distance between the centers O and O2 is 20 cm.
- PQ is a transverse common tangent that touches both circles and intersects OO2 at point X.
- We must find the length of O2X in centimetres.
- The tangent is such that perpendiculars from O and O2 to the tangent meet at right angles at the respective points of contact.

Concept / Approach:
When a transverse common tangent touches two circles, the line segment joining the centers intersects the tangent, creating two right triangles with the radii as one leg and segments OX and O2X as the other legs. These triangles are similar because they share an acute angle at X and both are right angled at the points of tangency. From similarity, the ratio of distances from X to the centers equals the ratio of the radii. This gives a proportional relationship between OX and O2X that can be combined with the known total distance OO2 to compute each segment.

Step-by-Step Solution:
Step 1: Let OX be the distance from O to X and O2X be the distance from O2 to X. Step 2: The total distance between the centers is OO2 = OX + O2X = 20 cm. Step 3: Consider the right triangle formed by center O, point X, and the point of tangency on the first circle. Its legs are OX (along OO2) and the radius 7 cm (perpendicular to the tangent). Step 4: Similarly, consider the right triangle formed by center O2, point X, and the point of tangency on the second circle. Its legs are O2X and radius 9 cm. Step 5: The two triangles share the angle at X and both have right angles at the points of tangency, so they are similar. Step 6: By similarity, the ratio of corresponding sides gives OX / O2X = 7 / 9. Step 7: Let OX = 7k and O2X = 9k for some positive constant k. Then OX + O2X = 7k + 9k = 16k = 20. Step 8: Solve for k: k = 20 / 16 = 5 / 4. Step 9: Compute O2X = 9k = 9 * (5 / 4) = 45 / 4 cm.

Verification / Alternative check:
We can also compute OX = 7k = 7 * (5 / 4) = 35 / 4 cm. Check that OX + O2X equals the total center distance: 35 / 4 + 45 / 4 = 80 / 4 = 20 cm, which matches the given value of OO2. This confirms that the proportional assignment based on similarity was correct and that the resulting segment lengths satisfy the required conditions.

Why Other Options Are Wrong:
Option 10: If O2X were 10 cm, then OX would be 10 cm as well, suggesting equal radii, which contradicts the given radii 7 cm and 9 cm and the similarity ratio.
Option 6: This value ignores the ratio 7 to 9 and does not satisfy OX + O2X = 20 when combined with a corresponding OX consistent with similarity.
Option 35/4: This is actually the length of OX, not O2X, so selecting it confuses the two segments.
Option 5: This small value for O2X would require OX to be 15 cm, but then OX / O2X would be 3, which does not match the radii ratio 7 / 9.

Common Pitfalls:
A common mistake is to assume the tangent is symmetric with respect to the midpoint of OO2, leading to equal segments OX and O2X. Another error is to apply the formula for the length of common tangents rather than using triangle similarity for the intersection point X. Carefully identifying the right triangles and using the ratio of radii to relate distances along OO2 is the correct method here.

Final Answer:
The length of segment O2X is 45/4 cm.