△ABC is drawn on a coordinate plane with vertices A(1,3) , B(6,6) , and C(3,1) and with medians indicating the midpoint of each of the lines AB , BC , and CA . Prove that the medians meet at a single point by finding the centroid. Express all results in fractions.(1 point) centroid= (,)
To find the centroid of a triangle, we need to find the average of the coordinates of its vertices.
Let's find the coordinates of the midpoint of AB first:
The coordinates of A are (1,3) and the coordinates of B are (6,6).
The midpoint of AB can be found by taking the average of the x-coordinates and the average of the y-coordinates:
Midpoint of AB: ( (1+6)/2 , (3+6)/2 ) = (7/2, 9/2)
Let's find the coordinates of the midpoint of BC:
The coordinates of B are (6,6) and the coordinates of C are (3,1).
The midpoint of BC can be found by taking the average of the x-coordinates and the average of the y-coordinates:
Midpoint of BC: ( (6+3)/2 , (6+1)/2 ) = (9/2, 7/2)
Finally, let's find the coordinates of the midpoint of CA:
The coordinates of C are (3,1) and the coordinates of A are (1,3).
The midpoint of CA can be found by taking the average of the x-coordinates and the average of the y-coordinates:
Midpoint of CA: ( (3+1)/2 , (1+3)/2 ) = (2, 2)
Now, we can find the centroid by taking the average of the coordinates of the midpoints:
Centroid: ( (7/2 + 9/2 + 2)/3 , (9/2 + 7/2 + 2)/3 )
Simplifying, we get:
Centroid: ( (7+9+2)/6 , (9+7+2)/6 )
Centroid: ( 18/6 , 18/6 )
Centroid: ( 3 , 3 )
Therefore, the centroid of triangle ABC is (3,3).