Two circles with diameters 15 cm and 13 cm intersect so that their common chord has length 12 cm. What is the distance between the centres of the circles (in cm)?

Introduction / Context:
This question involves intersecting circles and a common chord. The length of the common chord and the radii of the two circles are given, and we are asked to find the distance between the centres. This is a standard geometry configuration: the line joining the centres is perpendicular to the common chord and bisects it. Using right triangle relationships, we can relate the radii, the half chord length, and the distance from each centre to the chord to obtain the distance between centres.

Given Data / Assumptions:
• Diameter of the first circle is 15 cm, so its radius r₁ = 7.5 cm.
• Diameter of the second circle is 13 cm, so its radius r₂ = 6.5 cm.
• The length of the common chord is 12 cm.
• The chord is common to both circles and lies between their centres.
• The line joining the centres is perpendicular to the common chord and bisects it.
• We need the distance d between the two centres.

Concept / Approach:
If two circles intersect, their common chord is perpendicular to the line joining the centres and is bisected by that line. Let the midpoint of the chord be M. Then the distance from each centre to M, say d₁ and d₂, can be found from right triangles using Pythagoras theorem: d₁² = r₁² - (chord/2)² and d₂² = r₂² - (chord/2)². The distance between the centres is then d = d₁ + d₂ because the centres, the midpoint of the chord, and the chord itself form a straight configuration with M lying between the centres.

Step-by-Step Solution:
Step 1: Compute the radii: r₁ = 15 / 2 = 7.5 cm, r₂ = 13 / 2 = 6.5 cm. Step 2: The chord length is 12 cm, so half the chord length is 12 / 2 = 6 cm. Step 3: For the first circle, let d₁ be the distance from its centre to the chord midpoint. Use Pythagoras: d₁² = r₁² - 6². Step 4: Compute r₁² = 7.5² = 56.25, so d₁² = 56.25 - 36 = 20.25, giving d₁ = 4.5 cm. Step 5: For the second circle, let d₂ be the distance from its centre to the chord midpoint. Then d₂² = r₂² - 6². Step 6: Compute r₂² = 6.5² = 42.25, so d₂² = 42.25 - 36 = 6.25, giving d₂ = 2.5 cm. Step 7: The centres lie on opposite sides of the chord midpoint, so the distance between the centres is d = d₁ + d₂ = 4.5 + 2.5 = 7 cm.

Verification / Alternative check:
We can verify by visualising the configuration. The line of centres is a straight line that passes through the chord midpoint M, with distances d₁ and d₂ from the centres. Because the chord is common and perpendicular to this line, and its half length is 6 cm, the right triangles formed in each circle are consistent. Squaring the sums or recomputing the distances ensures there is no arithmetic mistake. Since both right triangles satisfy Pythagoras theorem with the same half chord, adding the distances 4.5 cm and 2.5 cm naturally gives the full separation between centres as 7 cm, which is a simple and plausible result.

Why Other Options Are Wrong:
3.5 cm is too small and would not allow a chord of length 12 cm to exist with the given radii.
7√2 cm and 14 cm are significantly larger distances, which would change the geometry so that either the circles do not intersect or the chord length changes.
5 cm does not match the sum of the distances d₁ and d₂ computed from the radii and chord length. Only 7 cm works with both radii and the 12 cm chord consistently.

Common Pitfalls:
Learners may forget that the chord is bisected by the line joining the centres and may incorrectly use the full chord length instead of half. Another common mistake is to subtract distances in the wrong order and attempt d = |d₁ - d₂| instead of d₁ + d₂. Errors in squaring the radii or half chord length can also lead to incorrect values for d₁ and d₂. Keeping track of the geometry and the positions of the centres relative to the chord is essential.

Final Answer:
The distance between the centres of the two circles is 7 centimetres.