AB is the diameter of a circle and has a length of 8 cm. From point A, a tangent to the circle is drawn, and point C lies on this tangent such that AC = 6 cm. What is the length, in centimetres, of segment BC joining B and C?

Introduction / Context:

This problem combines circle geometry, tangents, and the Pythagoras theorem. It reflects a standard pattern seen in many aptitude tests, where a diameter, a tangent, and a line joining a point on the circle to a point on a tangent form a right triangle.

Given Data / Assumptions:

• AB is the diameter of the circle and AB = 8 cm.
• Therefore, the radius of the circle is 4 cm.
• From point A on the circle, a tangent AC is drawn such that AC = 6 cm.
• Point B is the other endpoint of the diameter, and we need BC.
• The tangent at a point on the circle is perpendicular to the radius through that point.

Concept / Approach:

The key idea is to observe that AB is a diameter and the radius at A is along AB. The tangent at A is perpendicular to the radius, so AC is perpendicular to AB. Hence triangle ABC is a right triangle with the right angle at A, and we can apply the Pythagoras theorem to find BC.

Step-by-Step Solution:

Verification / Alternative check:

You can check the logic by noting the classic 6, 8, 10 Pythagorean triple. Whenever the legs are 6 and 8, the hypotenuse is 10. This confirms the calculation is consistent.

Why Other Options Are Wrong:

The values 12 cm, 8 cm, 7 cm, and 9 cm do not satisfy the relation BC^2 = 64 + 36. For example, 8^2 = 64 and 9^2 = 81 are too small, while 12^2 = 144 is too large, so none of them satisfy the Pythagoras theorem for this right triangle.

Common Pitfalls:

One common mistake is to assume a right angle at C instead of at A, or to forget that the tangent is perpendicular to the radius. Another error is to misidentify which side is the hypotenuse, leading to wrong substitution into the Pythagoras formula.

Final Answer:

The length of segment BC is 10 cm.