Triangle XYZ is drawn on a coordinate plane with vertices X(0,0) , Y(3,6) , and Z(4,3) and with medians indicating the midpoint of each line XY , YZ , and ZX . Prove that the medians meet at a single point by finding the centroid.
To find the centroid of triangle XYZ, we need to find the average of the coordinates of the three vertices. Let's call the coordinates of the centroid (a, b).
Since the midpoint of XY is (1.5, 3), the x-coordinate of the centroid is a = (0 + 1.5 + 4)/3 = 1.5.
Similarly, since the midpoint of YZ is (3.5, 4.5), the y-coordinate of the centroid is b = (0 + 6 + 4.5)/3 = 10.5/3 = 3.5.
Therefore, the centroid of triangle XYZ is (1.5, 3.5).