O is the circumcentre of triangle ABC. If the distance AO from vertex A to the circumcentre is 8 cm, what is the length of BO?

Introduction / Context:
This is a conceptual geometry question about the circumcentre of a triangle. The circumcentre is the point that is equidistant from all three vertices of the triangle, and it is the centre of the circumcircle passing through those vertices. The question checks your understanding of this fundamental property. Although no coordinates or side lengths are given, recognising the definition of the circumcentre is enough to answer correctly.

Given Data / Assumptions:

- Triangle ABC is any triangle (it may be acute, right or obtuse).
- O is the circumcentre of triangle ABC, that is, the centre of the circumcircle passing through A, B and C.
- AO, the distance from vertex A to the circumcentre, is 8 cm.
- We are asked to find BO, the distance from vertex B to the circumcentre.
- The standard property of a circumcentre applies: all vertices are equidistant from O.

Concept / Approach:
By definition, the circumcentre of a triangle is the point where the perpendicular bisectors of the sides meet, and it is equidistant from all three vertices. This means OA = OB = OC, where OA, OB and OC are radii of the circumcircle. Therefore, if you know the distance from the circumcentre to one vertex, you automatically know the distance to the other vertices, because all are equal to the circumradius.

Step-by-Step Solution:
Step 1: Recall that in any triangle, the circumcentre is the centre of a circle that passes through all three vertices A, B and C.Step 2: By this definition, distances OA, OB and OC are all equal and represent the radius r of the circumcircle.Step 3: The problem gives AO = 8 cm.Step 4: Therefore, OA = OB = OC = 8 cm.Step 5: Hence, BO must equal the same circumradius.Step 6: So, BO = 8 cm.Step 7: The length of BO is directly determined from the fundamental property of the circumcentre without any further calculation.

Verification / Alternative check:
You can imagine drawing any triangle and constructing its perpendicular bisectors, which meet at the circumcentre O. If you then draw the circle centred at O and passing through A, the radius OA fixes the size of the circle. Because B and C lie on the same circle, the distances OB and OC must also be equal to the radius. Changing the shape of the triangle while keeping the same circumcentre and circumradius does not alter this equality. This conceptual picture reconfirms that OB equals OA.

Why Other Options Are Wrong:
Values like 3 cm, 6 cm and 12 cm imply that different vertices are at different distances from O, which contradicts the definition of a circumcentre. The option Cannot be determined is also incorrect because the property OA = OB = OC is universally true for any triangle with a defined circumcentre, so no extra information such as side lengths or angles is needed to find BO once AO is known.

Common Pitfalls:
Some students confuse the circumcentre with other centres of a triangle such as the centroid, incenter or orthocentre. For these other centres, distances to the vertices are not necessarily equal, which may make them think that BO cannot be determined. Remember that only the circumcentre is equidistant from all vertices. Mixing up these centres is a common conceptual error in geometry problems.

Final Answer:
Therefore, the length of BO is 8 cm.