Triangle ABC
is drawn on a coordinate plane with vertices A(4,4)
, B(9,1)
, 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)
To find the centroid of a triangle, we need to find the average of the coordinates of the three vertices. Let's label the midpoint of AB as D, midpoint of BC as E, and midpoint of CA as F.
Finding the Midpoint of AB (D):
Using the midpoint formula:
x-coordinate of D = (x-coordinate of A + x-coordinate of B) / 2
= (4 + 9) / 2
= 13 / 2
y-coordinate of D = (y-coordinate of A + y-coordinate of B) / 2
= (4 + 1) / 2
= 5 / 2
Therefore, D = (13/2, 5/2).
Finding the Midpoint of BC (E):
Using the midpoint formula:
x-coordinate of E = (x-coordinate of B + x-coordinate of C) / 2
= (9 + 3) / 2
= 12 / 2
= 6
y-coordinate of E = (y-coordinate of B + y-coordinate of C) / 2
= (1 - 1) / 2
= 0 / 2
= 0
Therefore, E = (6, 0).
Finding the Midpoint of CA (F):
Using the midpoint formula:
x-coordinate of F = (x-coordinate of C + x-coordinate of A) / 2
= (3 + 4) / 2
= 7 / 2
y-coordinate of F = (y-coordinate of C + y-coordinate of A) / 2
= (-1 + 4) / 2
= 3 / 2
Therefore, F = (7/2, 3/2).
Now, we need to find the average of the x-coordinates and y-coordinates of points D, E, and F to find the centroid.
x-coordinate of centroid = (13/2 + 6 + 7/2) / 3
= (13/2 + 12/2 + 7/2) / 3
= (32/2) / 3
= 16/3
y-coordinate of centroid = (5/2 + 0 + 3/2) / 3
= (5/2 + 0/2 + 3/2) / 3
= (8/2) / 3
= 4/3.
Therefore, the centroid of triangle ABC is (16/3, 4/3), and the medians of the triangle meet at this point.
To find the centroid of a triangle, we need to find the average of the coordinates of its vertices. Let's calculate the coordinates of the centroid step by step:
1. Find the coordinates of the midpoint of line AB:
Midpoint of AB = [(x1 + x2) / 2, (y1 + y2) / 2]
Midpoint of AB = [(4 + 9) / 2, (4 + 1) / 2]
= [13 / 2, 5 / 2]
= (6.5, 2.5)
2. Find the coordinates of the midpoint of line BC:
Midpoint of BC = [(x1 + x2) / 2, (y1 + y2) / 2]
Midpoint of BC = [(9 + 3) / 2, (1 + (-1)) / 2]
= [12 / 2, 0 / 2]
= (6, 0)
3. Find the coordinates of the midpoint of line CA:
Midpoint of CA = [(x1 + x2) / 2, (y1 + y2) / 2]
Midpoint of CA = [(3 + 4) / 2, (-1 + 4) / 2]
= [7 / 2, 3 / 2]
= (3.5, 1.5)
4. Find the average of the x-coordinates of the midpoints:
Centroid x-coordinate = (6.5 + 6 + 3.5) / 3
= 16 / 3
= 5.33 (approximately)
5. Find the average of the y-coordinates of the midpoints:
Centroid y-coordinate = (2.5 + 0 + 1.5) / 3
= 4 / 3
= 1.33 (approximately)
Therefore, the coordinates of the centroid are approximately (5.33, 1.33). We have shown that the medians of the triangle meet at a single point, which is 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 of a triangle and is also known as the center of gravity or the center of mass.
To find the centroid, we first need to find the coordinates of each median's midpoint.
1. Find the midpoint of AB:
The coordinates of point A are (4,4) and the coordinates of point B are (9,1).
To find the midpoint, add the x-coordinates and divide by 2 to get the x-coordinate of the midpoint, and do the same for the y-coordinates to get the y-coordinate.
Midpoint of AB = ((4 + 9)/2, (4 + 1)/2) = (13/2, 5/2).
2. Find the midpoint of BC:
The coordinates of point B are (9,1) and the coordinates of point C are (3, -1).
Using the same approach as before, calculate the midpoint of BC.
Midpoint of BC = ((9 + 3)/2, (1 + -1)/2) = (12/2, 0) = (6, 0).
3. Find the midpoint of CA:
The coordinates of point C are (3, -1) and the coordinates of point A are (4,4).
Calculate the midpoint of CA.
Midpoint of CA = ((3 + 4)/2, (-1 + 4)/2) = (7/2, 3/2).
Now that we have the coordinates of the midpoints of AB, BC, and CA, we can find the centroid by taking the average of these points.
Centroid = ((13/2 + 6 + 7/2)/3, (5/2 + 0 + 3/2)/3)
= ((13/2 + 12/2 + 7/2)/3, (5/2 + 0 + 3/2)/3)
= (32/2/3, 8/2/3)
= (16/3, 4/3)
Therefore, the coordinates of the centroid of triangle ABC are (16/3, 4/3).