Understanding the Centroid
The centroid of a triangle is the point where the three medians of the triangle intersect. Think of it as the "balancing point" - if you were to cut a triangle out of cardboard, the centroid is where you could balance it on the tip of a pencil.
Centroid Formula
The centroid is calculated by finding the average of all the X-coordinates and Y-coordinates of the triangle's vertices:
[\text{Centroid X} = \frac{x_1 + x_2 + x_3}{3}]
[\text{Centroid Y} = \frac{y_1 + y_2 + y_3}{3}]
Where (x₁, y₁), (x₂, y₂), and (x₃, y₃) are the coordinates of the three vertices of the triangle.
Example Calculation
Let's find the centroid of a triangle with vertices at (2, 3), (4, 8), and (6, 2).
Step 1: Calculate the X-coordinate of the centroid:
[\text{Centroid X} = \frac{2 + 4 + 6}{3} = \frac{12}{3} = 4]
Step 2: Calculate the Y-coordinate of the centroid:
[\text{Centroid Y} = \frac{3 + 8 + 2}{3} = \frac{13}{3} \approx 4.33]
Result: The centroid is located at (4, 4.33).
Properties of Centroids
The centroid has several important properties:
- It always lies inside the triangle
- It divides each median in a 2:1 ratio
- It is the center of mass for a triangle with uniform density
- The centroid is always closer to the longest side of the triangle