Two equal circles each have radius 3 cm and the distance between their centres is 10 cm. What is the length of one transverse common tangent between the circles?

Introduction / Context:
This aptitude geometry question tests your understanding of common tangents to two circles, especially the formula for the length of a transverse common tangent. Problems like this are common in competitive exams because they combine circle geometry, Pythagorean thinking and remembering the difference between direct and transverse tangents. Knowing which formula applies in which configuration helps you quickly solve many similar tangent based questions.

Given Data / Assumptions:
• Two equal circles, each with radius r = 3 cm. • Distance between their centres d = 10 cm. • We are asked for the length of one transverse common tangent. • Circles are assumed to be in the same plane and non intersecting because d is greater than 2r.

Concept / Approach:
The key concept is that there are two types of common tangents to two circles: direct common tangents and transverse common tangents. For two circles with radii r1 and r2 and distance between centres d, the length of a direct common tangent is sqrt(d^2 - (r1 - r2)^2), while the length of a transverse common tangent is sqrt(d^2 - (r1 + r2)^2). Because the circles here are equal, r1 = r2 = r, which simplifies the expression. The required length can be found directly using the Pythagoras theorem in the right triangle formed by joining the centres and the tangent segment.

Step-by-Step Solution:
1. Let r1 = r2 = r = 3 cm and distance between centres d = 10 cm. 2. For a transverse common tangent, use the formula: length L = sqrt(d^2 - (r1 + r2)^2). 3. Compute r1 + r2 = 3 + 3 = 6 cm. 4. Compute d^2 = 10^2 = 100. 5. Compute (r1 + r2)^2 = 6^2 = 36. 6. Substitute into the formula: L = sqrt(100 - 36) = sqrt(64). 7. Therefore, L = 8 cm.

Verification / Alternative check:
One quick validation is to compare the value of the tangent length with the distance between centres and the radii. The sum of the radii is 6 cm and the distance between centres is 10 cm, so the tangent segment must be shorter than 10 cm but longer than 4 cm to form a valid right triangle. The computed length 8 cm clearly lies between 4 cm and 10 cm, which is consistent with a Pythagorean triple 6, 8, 10. This gives a strong confidence that the calculation is correct.

Why Other Options Are Wrong:
• 7 cm: This is slightly less than the correct value and does not satisfy the relation d^2 = L^2 + (r1 + r2)^2. • 9 cm: This is too large; if L = 9, then 9^2 + 6^2 would exceed 10^2, which is impossible. • 10 cm: The tangent length cannot equal the distance between centres; that would force the radii sum to be zero. • 6 cm: This would mean d^2 = 6^2 + 6^2 = 72, which is not equal to 100.

Common Pitfalls:
Many students confuse direct and transverse common tangents and mistakenly use sqrt(d^2 - (r1 - r2)^2) even when the question explicitly says transverse common tangent. Another frequent error is forgetting that the radii add for the transverse case, leading to an incorrect smaller value. Some learners also neglect units, but it is important to keep everything in centimetres and state the final answer with units.

Final Answer:
The length of one transverse common tangent between the two circles is 8 cm.