Two circles have radii 9 centimetres and 12 centimetres respectively, and the distance between their centres is 15 centimetres. The circles intersect in a common chord. What is the length, in centimetres, of this common chord?

Introduction / Context:
This question concerns the geometry of two intersecting circles. When two circles overlap, their intersection is a common chord. Given the radii of both circles and the distance between their centres, we can find the length of that chord using right triangle relationships. Such problems test spatial visualisation and familiarity with circle geometry and the Pythagoras theorem.

Given Data / Assumptions:

• Radius of the first circle R1 = 12 cm.
• Radius of the second circle R2 = 9 cm.
• Distance between centres of the circles d = 15 cm.
• The circles intersect, forming a common chord.
• We are asked to find the length of this chord in centimetres.
• All constructions are in a standard Euclidean plane.

Concept / Approach:
Consider line segment joining the two centres. The common chord is perpendicular to this line and passes through some point M between the centres. Let the half length of the chord be l. Then two right triangles form: one with hypotenuse R1 and another with hypotenuse R2, each having one leg l and the other legs being distances from centres to M. By using the relation between these distances and the fact that their sum is d, we can find l. The full chord length is then 2l.

Step-by-Step Solution:
Step 1: Let O1 and O2 be centres of radii 12 cm and 9 cm, and let M be the midpoint of the common chord. Step 2: Let O1M = a and O2M = b. Then a + b = d = 15 cm. Step 3: In right triangle O1M with the chord half length l, we have l^2 = R1^2 − a^2 = 12^2 − a^2. Step 4: In right triangle O2M, l^2 = R2^2 − b^2 = 9^2 − b^2. Step 5: Set these equal: 12^2 − a^2 = 9^2 − b^2, so 144 − a^2 = 81 − b^2. Step 6: Rearrange to get a^2 − b^2 = 144 − 81 = 63. Step 7: Factor the left side: (a − b)(a + b) = 63. But a + b = 15, so (a − b) * 15 = 63, giving a − b = 63 / 15 = 4.2. Step 8: Solve the system a + b = 15 and a − b = 4.2. Adding gives 2a = 19.2, so a = 9.6 cm, and b = 15 − 9.6 = 5.4 cm. Step 9: Use l^2 = R1^2 − a^2 = 144 − 9.6^2. Compute 9.6^2 = 92.16, so l^2 = 144 − 92.16 = 51.84. Step 10: Therefore l = √51.84 = 7.2 cm, so the full chord length = 2l = 14.4 cm.

Verification / Alternative check:
As a check, we can use the smaller circle. With b = 5.4 cm and R2 = 9 cm, l^2 should be 9^2 − 5.4^2 = 81 − 29.16 = 51.84, which again gives l = 7.2 cm and full chord 14.4 cm. Also, the distances a and b add to 15 cm, and the difference in squared radii equals (a − b)(a + b), confirming the internal consistency of the geometry and calculations.

Why Other Options Are Wrong:
6.8, 7.2 and 13.6 centimetres are either half lengths or intermediate values appearing in the computations, not the final chord length. For example, 7.2 cm is the half length l, not the full chord. The value 12 cm is a distractor based on one of the radii rather than on the actual intersection geometry. Only 14.4 cm matches 2 * 7.2 cm obtained from correct application of right triangle relations.

Common Pitfalls:
A common error is to treat the chord as if it belonged to only one circle and directly apply formulas without considering the second circle. Others may forget that the chord is perpendicular to the line joining centres at its midpoint, leading to incorrect triangles. Algebra mistakes in solving the system for a and b, or in squaring decimals, can also lead to wrong chord lengths. Careful use of the relations a + b = d and a^2 − b^2 = R1^2 − R2^2 keeps the calculation systematic.

Final Answer:
The length of the common chord is 14.4 cm.