Find the centroids of the triangles whose vertices are given below.(3, -5), (4, 3), (11, -4)

Explanation:-

The problem gives us three points in a 2-dimensional space, namely (3, -5), (4, 3), and (11, -4). These three points form a triangle. The problem asks us to find the centroid of this triangle.

The centroid of a triangle is the point where the three medians of the triangle intersect. A median of a triangle is a line segment connecting a vertex of the triangle to the midpoint of the opposite side. The centroid is the point where all three medians intersect, and it is also the center of mass of the triangle.

To find the centroid of the triangle formed by the given coordinates, we first need to find the midpoints of each side of the triangle. We can do this by averaging the x-coordinates and y-coordinates of the endpoints of each side. For example, the midpoint of the side connecting (3, -5) and (4, 3) can be found as follows:

[Midpoint_{AB} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \left(\frac{3 + 4}{2}, \frac{-5 + 3}{2}\right) = \left(\frac{7}{2}, -1\right)]

Similarly, we can find the midpoints of the other two sides of the triangle: