Find the Centroid of Each Triangle - Math Worksheet

To find the centroid of a triangle, we use a simple formula. The centroid is the "center of mass" or the balance point of the triangle.

The Formula:
If the three corners (vertices) of a triangle are at points $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, then the centroid $(C_x, C_y)$ is found by averaging the coordinates:

$$C_x = \frac{x_1 + x_2 + x_3}{3}$$
$$C_y = \frac{y_1 + y_2 + y_3}{3}$$

Basically, you add up all the x-values and divide by 3 to get the new x-coordinate. Then, you add up all the y-values and divide by 3 to get the new y-coordinate.

Let's solve each problem step-by-step.

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1) Triangle 1
* Identify Coordinates: Looking at the grid, the vertices are:
* $(-6, 0)$
* $(4, -5)$
* $(1, 2)$
* Calculate X: $\frac{-6 + 4 + 1}{3} = \frac{-1}{3}$
* Calculate Y: $\frac{0 + (-5) + 2}{3} = \frac{-3}{3} = -1$
* Centroid: $(-\frac{1}{3}, -1)$

2) Triangle 2
* Identify Coordinates:
* $(-4, 1)$
* $(3, 0)$
* $(0, -4)$
* Calculate X: $\frac{-4 + 3 + 0}{3} = \frac{-1}{3}$
* Calculate Y: $\frac{1 + 0 + (-4)}{3} = \frac{-3}{3} = -1$
* Centroid: $(-\frac{1}{3}, -1)$

3) Triangle 3
* Identify Coordinates:
* $(-5, 1)$
* $(-1, 0)$
* $(0, 5)$
* Calculate X: $\frac{-5 + (-1) + 0}{3} = \frac{-6}{3} = -2$
* Calculate Y: $\frac{1 + 0 + 5}{3} = \frac{6}{3} = 2$
* Centroid: $(-2, 2)$

4) Triangle 4
* Identify Coordinates:
* $(-3, -4)$
* $(4, -1)$
* $(2, 2)$
* Calculate X: $\frac{-3 + 4 + 2}{3} = \frac{3}{3} = 1$
* Calculate Y: $\frac{-4 + (-1) + 2}{3} = \frac{-3}{3} = -1$
* Centroid: $(1, -1)$

5) Triangle 5
* Identify Coordinates:
* $(-3, 1)$
* $(3, -4)$
* $(1, 2)$
* Calculate X: $\frac{-3 + 3 + 1}{3} = \frac{1}{3}$
* Calculate Y: $\frac{1 + (-4) + 2}{3} = \frac{-1}{3}$
* Centroid: $(\frac{1}{3}, -\frac{1}{3})$

6) Triangle 6
* Identify Coordinates:
* $(-5, -1)$
* $(3, 0)$
* $(5, 5)$
* Calculate X: $\frac{-5 + 3 + 5}{3} = \frac{3}{3} = 1$
* Calculate Y: $\frac{-1 + 0 + 5}{3} = \frac{4}{3}$
* Centroid: $(1, \frac{4}{3})$

7) Triangle 7
* Identify Coordinates:
* $(-5, -6)$
* $(5, 0)$
* $(-1, 1)$
* Calculate X: $\frac{-5 + 5 + (-1)}{3} = \frac{-1}{3}$
* Calculate Y: $\frac{-6 + 0 + 1}{3} = \frac{-5}{3}$
* Centroid: $(-\frac{1}{3}, -\frac{5}{3})$

8) Triangle 8
* Identify Coordinates:
* $(-2, 0)$
* $(5, 2)$
* $(2, -5)$
* Calculate X: $\frac{-2 + 5 + 2}{3} = \frac{5}{3}$
* Calculate Y: $\frac{0 + 2 + (-5)}{3} = \frac{-3}{3} = -1$
* Centroid: $(\frac{5}{3}, -1)$

Final Answer:
1) $(-\frac{1}{3}, -1)$
2) $(-\frac{1}{3}, -1)$
3) $(-2, 2)$
4) $(1, -1)$
5) $(\frac{1}{3}, -\frac{1}{3})$
6) $(1, \frac{4}{3})$
7) $(-\frac{1}{3}, -\frac{5}{3})$
8) $(\frac{5}{3}, -1)$