Triangle ABC is drawn on a coordinate plane with vertices A(−3,−3) , B(0,6) , and C(4,−3) and medians indicating the midpoint of each of the lines AB , BC , and CA . Prove the medians meet at a single point by finding the centroid.(1 point) Responses (16,3) left parenthesis Start Fraction 1 over 6 End Fraction comma 3 right parenthesis (0,13) left parenthesis 0 comma Start Fraction 1 over 3 End Fraction right parenthesis (13,1) left parenthesis Start Fraction 1 over 3 End Fraction comma 1 right parenthesis (13,0)
To find the centroid, we need to find the average of the coordinates of the vertices.
Let's first find the coordinates of the midpoint of AB. The x-coordinate is the average of the x-coordinates of A and B, which is:
(-3 + 0) / 2 = -1.5
The y-coordinate is the average of the y-coordinates of A and B, which is:
(-3 + 6) / 2 = 1.5
So the midpoint of AB is (-1.5, 1.5).
Similarly, we can find the midpoint of BC:
(0 + 4) / 2 = 2
(-3 + (-3)) / 2 = -3
So the midpoint of BC is (2, -3).
Lastly, we find the midpoint of CA:
(-3 + 4) / 2 = 0.5
(-3 + (-3)) / 2 = -3
So the midpoint of CA is (0.5, -3).
Now, to find the centroid, we take the average of the x-coordinates and the average of the y-coordinates.
The x-coordinate is:
(-1.5 + 2 + 0.5) / 3 = 1 / 6 ≈ 0.17
The y-coordinate is:
(1.5 + (-3) + (-3)) / 3 = -5 / 3 ≈ -1.67
So the centroid is approximately (0.17, -1.67).
Therefore, the medians meet at the centroid, which is approximately (0.17, -1.67).