Question

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.centroid= ( , )

Answer 1

The centroid of triangle XYZ with vertices X(0,0), Y(3,6), and Z(4,3) is located at C(2, 2), found by calculating the midpoints of the triangle's sides and the intersection of its medians.

Step-by-step explanation:

The centroid of a triangle is the point where its three medians intersect; it is also the center of mass of the triangle. To find the centroid, we calculate the midpoint of each side, and then find the point of intersection of the medians. Let's find the midpoints of the sides of triangle XYZ with vertices X(0,0), Y(3,6), and Z(4,3).

• The midpoint of XY is M1((0+3)/2, (0+6)/2) = M1(1.5, 3).
• The midpoint of YZ is M2((3+4)/2, (6+3)/2) = M2(3.5, 4.5).
• The midpoint of ZX is M3((4+0)/2, (3+0)/2) = M3(2, 1.5).

The median from vertex X to midpoint M2 can be represented by the equation of the line passing through X(0,0) and M2(3.5, 4.5). Similarly, the median from vertex Y to midpoint M3 can be represented by the line passing through Y(3,6) and M3(2, 1.5). These lines will intersect at the centroid of the triangle. By solving the system of equations representing these two lines, we find the intersection point, which is the centroid of triangle XYZ. This calculation will reveal that the centroid is at C(2, 2), which is the average of the x-coordinates (0+3+4)/3 and y-coordinates (0+6+3)/3 of the vertices of triangle XYZ.