The centroid of the triangle with vertices (3, −5), (−7, 4) and (10, −2) is (2, k). Find k.

Class 10Coordinate GeometryGeometry — Coordinates

Step-by-Step Solution

The centroid's

y

-coordinate is the average of the three

y

-values:

k = \frac{- 5 + 4 + ( - 2 )}{3} = \frac{- 3}{3} = - 1

. The

x

-coordinate confirms the data:

\frac{3 + ( - 7 ) + 10}{3} = \frac{6}{3} = 2

✓. Quoting the sum −3 without dividing by 3 is the usual slip, and dropping the minus sign gives 1.

Answer: C) -1

Key idea: Centroid of a triangle

The **centroid** is the point where the three medians of a triangle meet (a median connects a vertex to the midpoint of the opposite side). It is the 'balance point' of the triangle.

**Coordinates**: just the average of the three vertices:

G = (\frac{x _{1} + x _{2} + x _{3}}{3}, \frac{y _{1} + y _{2} + y _{3}}{3})

.

**Example**: Triangle with A(0, 0), B(6, 0), C(3, 9).
Centroid:

(\frac{0 + 6 + 3}{3}, \frac{0 + 0 + 9}{3})

= (3, 3).

A useful invariant — the centroid divides each median in ratio 2:1 from the vertex.

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