Triangle ABC is drawn on a coordinate plane with vertices A(−2,−3) , B(4, 0) , and C(−2, 2) and medians indicating the midpoint of each of the line segments AB¯¯¯¯¯¯¯¯ , BC¯¯¯¯¯¯¯¯ , and CA¯¯¯¯¯¯¯¯ . Prove that the medians meet at a single point by finding the centroid.
To prove that the medians of triangle ABC meet at a single point, we need to find the coordinates of the centroid. The centroid is the point of intersection of the medians and divides each median into two segments in a 2:1 ratio.
To find the centroid, we first find the coordinates of the midpoint of each line segment. The midpoint of AB is the average of the coordinates of A and B:
Midpoint of AB = ((-2 + 4)/2, (-3 + 0)/2) = (1, -3/2)
Similarly, the midpoint of BC is the average of the coordinates of B and C:
Midpoint of BC = ((4 + (-2))/2, (0 + 2)/2) = (1, 1)
Finally, the midpoint of CA is the average of the coordinates of C and A:
Midpoint of CA = ((-2 + (-2))/2, (2 + (-3))/2) = (-2, -1/2)
Now we have the coordinates of the midpoints of all three line segments. Next, we find the coordinates of the centroid by taking the average of the coordinates of the midpoints:
Centroid = ((1 + 1 + (-2))/3, (-3/2 + 1 - 1/2)/3)
= (0, -1/6)
Therefore, the centroid of triangle ABC is the point (0, -1/6).
Since the centroid is the point of intersection of the medians, we have proven that the medians of triangle ABC meet at a single point.
(0,−13)
open paren 0 comma negative 1 third close paren
(0, 53)
open paren 0 comma 5 thirds close paren
(0, 0)
open paren 0 comma 0 close paren
(−1, 53)
these are one of the answers
I apologize for the incorrect calculation in my previous response.
To find the centroid of triangle ABC, we need to take the average of the coordinates of the midpoints.
The midpoint of AB is ((-2+4)/2, (-3+0)/2) = (1, -3/2)
The midpoint of BC is ((4+(-2))/2, (0+2)/2) = (1, 1)
The midpoint of CA is ((-2+(-2))/2, (2+(-3))/2) = (-2, -1/2)
Now, to find the centroid, we take the average of these midpoints:
Centroid = ((1+1+(-2))/3, (-3/2+1-1/2)/3) = (0, -1/6).
Therefore, the centroid of triangle ABC is (0, -1/6).
This means that the medians of triangle ABC do indeed meet at a single point, which is the centroid with coordinates (0, -1/6).
that is not the answer
Apologies for the confusion in my previous response. Let's re-calculate to find the correct coordinates of the centroid of triangle ABC.
The midpoint of AB is ((-2+4)/2, (-3+0)/2) = (1, -3/2).
The midpoint of BC is ((4+(-2))/2, (0+2)/2) = (1, 1).
The midpoint of CA is ((-2+(-2))/2, (2+(-3))/2) = (-2, -1/2).
To find the centroid, we need to find the average of these midpoints:
Centroid = ((1+1+(-2))/3, (-3/2+1-1/2)/3) = (0, -1/6).
Therefore, the centroid of triangle ABC is (0, -1/6).
If the calculated coordinates of the centroid differ from your answer, please provide the correct answer, and I will re-evaluate.
these are the answers provided.
(0,−1/3)
(0, 5/3)
(0, 0)
(−1, 5/3)
Apologies for the confusion. It seems I made an error in my previous responses. Let's re-calculate to find the correct coordinates of the centroid of triangle ABC.
The midpoint of AB is ((-2+4)/2, (-3+0)/2) = (1, -3/2).
The midpoint of BC is ((4+(-2))/2, (0+2)/2) = (1, 1).
The midpoint of CA is ((-2+(-2))/2, (2+(-3))/2) = (-2, -1/2).
To find the centroid, we need to find the average of these midpoints:
Centroid = ((1+1+(-2))/3, (-3/2+1-1/2)/3) = (0, -1/3).
Therefore, the centroid of triangle ABC is (0, -1/3).
My previous responses were incorrect. I apologize for the confusion. The correct coordinates of the centroid are indeed (0, -1/3).
Thank you for providing the correct answers.