In triangle ABC, AD is a median drawn to side BC and O is the centroid of the triangle. If AO = 10 cm, what is the length of OD (in centimetres)?

Introduction / Context:
This question is about medians and the centroid of a triangle. Understanding how the centroid divides each median in a fixed ratio is a standard result in coordinate geometry and Euclidean geometry, and it appears frequently in exam problems on triangle properties.

Given Data / Assumptions:

• Triangle ABC has median AD drawn from vertex A to side BC, so D is the midpoint of BC.
• O is the centroid of triangle ABC (point of intersection of all three medians).
• The length AO from vertex A to centroid O is 10 cm.
• We need the length of segment OD from centroid O to midpoint D.

Concept / Approach:
A key property of the centroid is that it divides every median of a triangle in the ratio 2 : 1, measured from the vertex to the opposite side. This means that for median AD, AO : OD = 2 : 1. If AO is known, we can express the full median length AD in terms of AO and then find OD as the remaining one third of the median. No angles or coordinates are required if this property is remembered.

Step-by-Step Solution:
Step 1: Use the centroid property: AO : OD = 2 : 1 along the median AD. Step 2: Let AD be the full length of the median. Then AO = (2/3) * AD and OD = (1/3) * AD. Step 3: Given AO = 10 cm, substitute: (2/3) * AD = 10. Step 4: Solve for AD: AD = (10 * 3) / 2 = 30 / 2 = 15 cm. Step 5: Now find OD: OD = (1/3) * AD = (1/3) * 15 = 5 cm.

Verification / Alternative Check:
We can also reason directly from the ratio AO : OD = 2 : 1. If AO is 10 cm and represents two equal parts, then each part is 5 cm. Therefore OD, which is one such part, must be 5 cm. This matches the calculation using the median length and confirms the result.

Why Other Options Are Wrong:
Option b, 20 cm, would make OD longer than AO, contradicting the 2 : 1 ratio from vertex to midpoint. Option c, 10 cm, would imply AO and OD are equal, which is again inconsistent with the centroid property. Option d, 30 cm, is even larger and clearly impossible because the entire median would need to be longer than 45 cm in that case, which does not match AO = 10 cm.

Common Pitfalls:
A common mistake is to invert the ratio and assume AO is one third of the median instead of two thirds. Another pitfall is to think that the centroid divides the medians in a 1 : 1 ratio like a midpoint, which is not correct. Remember that the centroid is closer to the base than to the vertex and always lies two thirds of the way from each vertex along its median.

Final Answer:
The length of OD is 5 cm.