Two circles touch each other externally. The distance between their centres is 7 cm. If the radius of one circle is 4 cm, what is the radius of the other circle (in centimetres)?

Introduction / Context:
This question tests the basic geometric property of two circles that touch each other externally. It is a straightforward application of the relation between the distance of the centres and the radii of the two circles.

Given Data / Assumptions:

• Two circles touch each other externally.
• The distance between their centres is 7 cm.
• The radius of one circle is 4 cm.
• We must find the radius of the other circle.

Concept / Approach:
When two circles touch externally, the distance between their centres is equal to the sum of their radii. If the radii are r₁ and r₂, and d is the distance between the centres, then for external tangency we have d = r₁ + r₂. Here d is given as 7 cm, and one radius is known to be 4 cm. We simply rearrange the equation to solve for the unknown radius.

Step-by-Step Solution:
Step 1: Let the unknown radius be r cm. Step 2: Use the external tangency relation: distance between centres = sum of radii. Step 3: Substitute the known values: 7 = 4 + r. Step 4: Solve for r: r = 7 − 4 = 3. Step 5: Thus the radius of the other circle is 3 cm.

Verification / Alternative Check:
Check by recomputing the distance using the found radius. If one circle has radius 4 cm and the other has radius 3 cm, then r₁ + r₂ = 4 + 3 = 7 cm, which matches the given distance between centres. This confirms that the two circles would just touch externally with no overlap and no gap.

Why Other Options Are Wrong:
A radius of 4 cm for the other circle would give 4 + 4 = 8 cm between centres, not 7 cm. A radius of 5.5 cm or 3.5 cm would yield distances 9.5 cm or 7.5 cm, which also do not match the given distance. Only 3 cm satisfies the condition of external tangency with the given centre separation.

Common Pitfalls:
Sometimes learners mix up formulas for external and internal tangency. For internal tangency (one circle inside another), the distance between centres is the difference of the radii, not the sum. In this question, the circles touch externally, so it must be the sum. Carefully noting the word externally helps to apply the correct relation.

Final Answer:
The radius of the other circle is 3 cm.