Prove that when the three medians of the triangle are drawn, they meet at a single point. What is the point of the centroid? A-5,1 B1,7 C 1, -5
To prove that the three medians of a triangle meet at a single point, we can use the property that the medians divide each other in a 2:1 ratio. Let's call this point of intersection as G, and the three vertices of the triangle as A, B, and C.
Let M be the midpoint of side BC. Since G divides AM in a 2:1 ratio, we can write:
AG/AM = 2/1
Similarly, if N is the midpoint of side AC, we have:
BG/BN = 2/1
And if P is the midpoint of side AB, we have:
CG/CP = 2/1
We can rewrite these three equations as:
AG = 2/3 * AM
BG = 2/3 * BN
CG = 2/3 * CP
Now, let's add these three equations:
AG + BG + CG = 2/3 * (AM + BN + CP)
Since AM + BN + CP is equal to the perimeter of the triangle (AB + BC + AC), we can write:
AG + BG + CG = 2/3 * (AB + BC + AC)
This equation shows that the sum of the three segments AG, BG, and CG is equal to 2/3 of the perimeter of the triangle.
Now, let's look at another property of the centroid. The centroid divides each median in a 2:1 ratio. So we can write:
GM/AG = 1/2
GM/BG = 1/2
GM/CG = 1/2
Adding these three equations, we get:
GM + GM + GM = AG/2 + BG/2 + CG/2
3GM = 1/2 (AG + BG + CG)
Since AG + BG + CG is equal to 2/3 of the perimeter of the triangle, we have:
3GM = 2/2 * 2/3 (perimeter)
3GM = 4/3 (perimeter)
GM = 4/9 (perimeter)
This equation shows that each segment GM is equal to 4/9 of the perimeter of the triangle.
Now, if we consider the fact that the sum of the three segments AG, BG, and CG is equal to 2/3 of the perimeter of the triangle, we can conclude that the segment GM is equal to 1/3 of the sum of AG, BG, and CG.
Therefore, the point G is the point of intersection of the three medians, and it is located 1/3 of the distance from each vertex to the midpoint of the opposite side.
To determine the coordinates of the centroid, we need to calculate the average of the x-coordinates and the average of the y-coordinates of the three vertices of the triangle.
Given the vertices A(5,1), B(1,7), and C(1,-5):
Average x-coordinate = (5 + 1 + 1)/3 = 7/3
Average y-coordinate = (1 + 7 - 5)/3 = 3/3 = 1
Therefore, the centroid of the triangle is located at the point (7/3, 1).
Answer: The point of the centroid is (7/3, 1).