Two circles touch each other internally. The radius of the smaller circle is 6 centimetres and the distance between the centres of the two circles is 3 centimetres. Find the radius of the larger circle in centimetres.

Introduction / Context:
This question deals with two circles that touch each other internally. It tests your knowledge of the relationship between the radii of the two circles and the distance between their centres in the case of internal tangency. Such problems often appear in coordinate geometry and circle-based aptitude questions.

Given Data / Assumptions:

• There are two circles that touch each other internally.
• The radius of the smaller circle is r_small = 6 cm.
• The distance between the centres of the two circles is d = 3 cm.
• We must find the radius of the larger circle, r_large.
• The point of contact lies on the line joining the centres.

Concept / Approach:
For two circles that touch internally, the distance between their centres is equal to the difference of their radii:
d = r_large − r_small.
This is because, for internal tangency, the smaller circle lies inside the larger one and the common tangent point lies between the two centres. Using this simple linear relationship, we can solve for r_large by adding d and r_small.

Step-by-Step Solution:
Step 1: Recall the formula for internal tangency: d = r_large − r_small. Step 2: Substitute the known values: d = 3 cm and r_small = 6 cm. Step 3: So 3 = r_large − 6. Step 4: Add 6 to both sides: r_large = 3 + 6 = 9 cm. Step 5: Therefore, the radius of the larger circle is 9 cm.

Verification / Alternative check:
We can check if the geometry makes sense. If r_large = 9 cm and r_small = 6 cm, then the difference r_large − r_small = 9 − 6 = 3 cm, which matches the given distance between centres. This is consistent with internal tangency, where the smaller circle sits inside the larger one and touches it at exactly one point along the line connecting the centres.

Why Other Options Are Wrong:
7.5 cm and 8 cm: These values give differences of 1.5 cm and 2 cm between radii, which do not match the given distance of 3 cm.
10 cm: This would give a difference of 4 cm, again inconsistent with the distance of 3 cm.
6 cm: This would mean both circles have the same radius and cannot have centres 3 cm apart and touch internally in the way described.

Common Pitfalls:
Students sometimes mistakenly use the formula for external tangency, where the centre distance equals the sum of the radii, d = r_large + r_small. Here, because one circle is inside the other, the correct relationship is the difference of radii. Carefully identifying whether circles touch internally or externally is crucial before applying the appropriate formula.

Final Answer:
The radius of the larger circle is 9 cm.