A median of a triangle has a total length of 12 units. How far is the centroid from the vertex, and how far is it from the midpoint of the opposite side?
A6 units from the vertex, 6 units from the midpoint
B4 units from the vertex, 8 units from the midpoint
C8 units from the vertex, 4 units from the midpoint
D3 units from the vertex, 9 units from the midpoint
The centroid divides each median in a 2:1 ratio measured FROM THE VERTEX. So the centroid is 2/3 of the total length from the vertex — (2/3)×12 = 8 — and 1/3 from the midpoint — (1/3)×12 = 4. Option A confuses this with a 1:1 split (the midpoint of the median), which is wrong. The centroid is closer to the midpoint side, not the vertex side.
Question 2 Multiple Choice
A triangle has vertices at A(0, 0), B(6, 0), and C(0, 6). What are the coordinates of the centroid?
A(3, 3)
B(2, 2)
C(6, 6)
D(1, 1)
The centroid coordinates are the average of the three vertices: ((0+6+0)/3, (0+0+6)/3) = (6/3, 6/3) = (2, 2). Option A is a common error — students sometimes find the midpoint of one side rather than averaging all three vertices. The formula G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3) always gives the correct centroid.
Question 3 True / False
The centroid of a triangle is located one-third of the way from the vertex to the midpoint of the opposite side.
TTrue
FFalse
Answer: False
This reverses the ratio. The centroid is TWO-THIRDS of the way from the vertex to the midpoint — not one-third. It is one-third from the midpoint side and two-thirds from the vertex. A way to remember: the centroid is the balance point, and it sits closer to the 'heavy' base than to the vertex tip.
Question 4 True / False
For any triangle — whether acute, obtuse, or right — the centroid always lies inside the triangle.
TTrue
FFalse
Answer: True
Unlike the circumcenter (which can fall outside an obtuse triangle) and the orthocenter (which falls outside obtuse triangles), the centroid is always inside the triangle. This follows from the fact that it is the average of the three vertices — an average of three points always lies within the convex hull of those points, and a triangle is convex.
Question 5 Short Answer
Why is the centroid called the 'center of mass' or 'balance point' of a triangle, and how does this physical meaning connect to the coordinate formula ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)?
Think about your answer, then reveal below.
Model answer: If a triangle is cut from a uniform material, every point has equal mass per unit area. The center of mass is the weighted average of all those mass positions. Because the triangle is uniform, this reduces to the simple arithmetic average of the three vertices' coordinates — (x₁+x₂+x₃)/3 and (y₁+y₂+y₃)/3. This is exactly the centroid formula, so the geometric centroid and the physical balance point coincide: you could balance the triangular cutout on a pin placed exactly at this coordinate.
The connection reveals why the formula works: averaging the vertex coordinates is equivalent to finding the center of mass of a uniform triangle. This also explains the 2:1 ratio — the centroid sits 2/3 of the way from each vertex because the 'mass' of the opposite half of the triangle pulls it toward the base.