In a circle of radius 8 cm, AB and AC are two equal chords, each measuring 12 cm. If B and C are the other endpoints of these chords on the circle, what is the length of the chord BC that joins points B and C?

Introduction / Context:
In this geometry question, we work with chords in a circle and use basic circle properties to find the length of an unknown chord. The circle has radius 8 cm, and two equal chords AB and AC measure 12 cm each. Points B and C lie on the circle, and we are asked to determine the length of the chord BC. This tests understanding of the relation between chord length and the central angle that subtends the chord, as well as some simple trigonometry inside a circle.

Given Data / Assumptions:
- Radius of the circle r = 8 cm. - Chords AB and AC are equal: AB = AC = 12 cm. - B and C lie on the circle, and BC is the chord joining them. - The centre of the circle is denoted by O. - Standard chord length formula is used: chord length = 2 * r * sin(central angle / 2).

Concept / Approach:
To solve this, we use the fact that equal chords subtend equal central angles at the centre of a circle. Let the central angle subtended by chord AB be θ. Then chord AB has length 2 * r * sin(θ / 2). From AB = 12 cm and r = 8 cm, we can compute sin(θ / 2). Once we know θ, we can find the central angle between B and C, which is the angle subtending chord BC. Because AB and AC are symmetric around OA, if OB and OC are drawn, the angle BOC becomes 2θ. Then we again use the chord formula for BC with angle BOC to get its length.

Step-by-Step Solution:
Step 1: Let O be the centre of the circle. Draw OA, OB, and OC so that OA, OB, and OC are radii of length 8 cm. Step 2: Let the central angle AOB be θ. Since AB is a chord of length 12 cm, use AB = 2 * r * sin(θ / 2). Step 3: Substitute AB = 12 and r = 8: 12 = 2 * 8 * sin(θ / 2) which simplifies to 12 = 16 * sin(θ / 2). Step 4: Therefore sin(θ / 2) = 12 / 16 = 3 / 4. Step 5: Compute cos(θ / 2) using the identity sin^2(x) + cos^2(x) = 1. So cos(θ / 2) = √(1 - (3 / 4)^2) = √(1 - 9 / 16) = √(7 / 16) = √7 / 4. Step 6: Now find sin(θ) using the double angle formula sin(θ) = 2 * sin(θ / 2) * cos(θ / 2) = 2 * (3 / 4) * (√7 / 4) = (3 * √7) / 8. Step 7: Points B and C are symmetric with respect to OA, so the central angle BOC is 2θ. Step 8: Chord BC subtends angle BOC = 2θ at the centre. The chord length formula uses half of this angle, so BC = 2 * r * sin(BOC / 2) = 2 * r * sin(θ). Step 9: Substitute r = 8 and sin(θ) = (3 * √7) / 8 to get BC = 2 * 8 * (3 * √7 / 8) = 16 * (3 * √7 / 8) = 6√7 cm.

Verification / Alternative check:
An alternative method is to construct triangle OAB and use right triangle geometry by dropping a perpendicular from O to chord AB, then repeating a similar idea for BC. However, the trigonometric method used above is standard and efficient. Numeric substitution gives a positive value for BC, and the units are consistent in centimetres, so the result is reasonable.

Why Other Options Are Wrong:
Option 2√6 cm is too small and would correspond to a much smaller central angle than the one created by points B and C. Option 3√6 cm also does not match the value of BC found using the correct chord and angle relations. Option 3√7 cm is close in form but is only half of the correct value 6√7 cm.

Common Pitfalls:
Many learners forget that the chord formula uses half the central angle, leading to mistakes when relating chord length to the angle. Others may assume BC is equal to AB simply because AB and AC are equal, which is not true. It is also easy to confuse angle θ with 2θ when moving from chords AB and AC to chord BC. Careful use of the double angle identity for sine is important here.

Final Answer:
The length of chord BC in the given circle is 6√7 cm.