△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.
To prove that the medians of triangle ABC meet at a single point, we need to find the centroid of the triangle. The centroid is the point of intersection of the medians.
To find the coordinates of the centroid, we need to find the midpoints of the sides of triangle ABC.
Let's start by finding the coordinates of the midpoint of side AB.
The coordinates of point A are (1, 3), and the coordinates of point B are (6, 6). To find the midpoint, we add the x-coordinates and divide by 2, and add the y-coordinates and divide by 2:
Midpoint of AB:
x-coordinate = (1 + 6) / 2 = 7 / 2
y-coordinate = (3 + 6) / 2 = 9 / 2
So, the midpoint of AB is (7/2, 9/2).
Next, let's find the coordinates of the midpoint of side BC.
The coordinates of point B are (6, 6), and the coordinates of point C are (3, 1). Following the same process as before:
Midpoint of BC:
x-coordinate = (6 + 3) / 2 = 9 / 2
y-coordinate = (6 + 1) / 2 = 7 / 2
Therefore, the midpoint of BC is (9/2, 7/2).
Finally, let's find the coordinates of the midpoint of side CA.
The coordinates of point C are (3, 1), and the coordinates of point A are (1, 3). Again, following the same process:
Midpoint of CA:
x-coordinate = (3 + 1) / 2 = 4 / 2 = 2
y-coordinate = (1 + 3) / 2 = 4 / 2 = 2
So, the midpoint of CA is (2, 2).
Now that we have the coordinates of the midpoints of the sides of triangle ABC, we can find the centroid.
The centroid is the point of intersection of the medians, which can be calculated as the average of the x-coordinates and the average of the y-coordinates of the midpoints.
Centroid:
x-coordinate = (7/2 + 9/2 + 2) / 3 = (7 + 9 + 6) / 6 = 22 / 6 = 11 / 3
y-coordinate = (9/2 + 7/2 + 2) / 3 = (9 + 7 + 6) / 6 = 22 / 6 = 11 / 3
The centroid of triangle ABC is thus (11/3, 11/3).
Therefore, we have proven that the medians of triangle ABC meet at a single point, which is the centroid with coordinates (11/3, 11/3).