Two circles touch externally, so the distance between their centres equals the sum of their radii. The distance between their centres is 12 cm. The radius of one circle is 7 cm. Find the diameter of the other circle (in cm).

Introduction / Context:
This question tests the basic property of externally tangent circles. When two circles touch externally, they meet at exactly one point, and the line joining their centres passes through that point of tangency. In this case, the distance between their centres equals the sum of their radii: d = r1 + r2. If you know the centre distance and one radius, you can find the other radius by subtraction: r2 = d - r1. Finally, the diameter of the second circle is 2*r2. This is a direct substitution and arithmetic problem with a clear geometry meaning.

Given Data / Assumptions:

• Circles touch externally
• Distance between centres d = 12 cm
• Radius of one circle r1 = 7 cm
• For external touching: d = r1 + r2
• Diameter of other circle = 2*r2

Concept / Approach:
Use d = r1 + r2 to find r2. Then compute diameter = 2*r2.

Step-by-Step Solution:
d = r1 + r2 12 = 7 + r2 r2 = 12 - 7 = 5 cm Diameter of other circle = 2*r2 = 2*5 = 10 cm

Verification / Alternative check:
If the other circle’s radius were 5 cm, then the sum of radii would be 7 + 5 = 12 cm, exactly matching the given centre distance. This is precisely the condition for external tangency, so the computed radius and diameter are consistent with the geometry.

Why Other Options Are Wrong:
5 is the radius, not the diameter. 8 and 12 imply radii 4 and 6, which would not sum with 7 to make 12. 14 is the diameter of the first circle (2*7), not the second.

Common Pitfalls:
Adding instead of subtracting to get r2, confusing radius with diameter, or mixing up internal vs external tangency rules (difference of radii vs sum of radii).

Final Answer:
The diameter of the other circle is 10 cm.