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.
To prove that the medians of triangle ABC meet at a single point (the centroid), we can find the coordinates of the midpoint of each side and then find the intersection point of these medians.
Let's find the coordinates of the midpoints first:
Midpoint of AB:
x-coordinate = (x1 + x2) / 2 = (-3 + 0) / 2 = -3/2
y-coordinate = (y1 + y2) / 2 = (-3 + 6) / 2 = 3/2
So, the midpoint of AB is (-3/2, 3/2).
Midpoint of BC:
x-coordinate = (x2 + x3) / 2 = (0 + 4) / 2 = 2
y-coordinate = (y2 + y3) / 2 = (6 + -3) / 2 = 3/2
So, the midpoint of BC is (2, 3/2).
Midpoint of CA:
x-coordinate = (x3 + x1) / 2 = (4 + -3) / 2 = 1/2
y-coordinate = (y3 + y1) / 2 = (-3 + -3) / 2 = -3
So, the midpoint of CA is (1/2, -3).
Now, let's find the equation of the line passing through the midpoints of AB and BC:
Slope of the line passing through the midpoints of AB and BC = (3/2 - 3/2) / (-3/2 - 2) = 0 / (-3/2 - 4/2) = 0 / (-7/2) = 0
Since the slope is 0, the line is horizontal.
The equation of the line passing through the midpoint of AB and BC is y = 3/2.
Next, let's find the equation of the line passing through the midpoints of BC and CA:
Slope of the line passing through the midpoints of BC and CA = (3/2 - -3) / (2 - 1/2) = (3/2 + 3) / (4/2 - 1/2) = (9/2) / (3/2) = 9/3 = 3
So, the equation of the line passing through the midpoints of BC and CA is y = 3x - 3/2.
To find the intersection point of these two lines, we can solve the system of equations y = 3/2 and y = 3x - 3/2:
3/2 = 3x - 3/2
4/2 = 3x
2 = 3x
x = 2/3
Substituting this value of x back into either equation, we find y = 3/2.
Therefore, the intersection point of the medians is (2/3, 3/2), which is the centroid of triangle ABC.
Hence, we have proved that the medians of triangle ABC meet at a single point, which is the centroid.
the answers provided are
(0,1/3)
(1/6,3)
(1/3,0)
(1/3,1)
Apologies for the error in the previous response. Let's recalculate the coordinates of the centroid.
To find the centroid, we need to find the average of the coordinates of the vertices of the triangle.
Let's find the x-coordinate of the centroid:
x-coordinate = (x1 + x2 + x3) / 3 = (-3 + 0 + 4) / 3 = 1/3
Now, let's find the y-coordinate of the centroid:
y-coordinate = (y1 + y2 + y3) / 3 = (-3 + 6 - 3) / 3 = 0
Therefore, the coordinates of the centroid are (1/3, 0).
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 concurrency of the medians.
To find the centroid, we can use the following steps:
Step 1: Find the midpoints of the sides of the triangle.
The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) can be found using the midpoint formula: ((x1+x2)/2, (y1+y2)/2).
Using this formula, we can find the midpoints of the sides AB, BC, and CA:
Midpoint of AB:
x = (-3 + 0) / 2 = -1.5
y = (-3 + 6) / 2 = 1.5
So the midpoint of AB is M(-1.5, 1.5).
Midpoint of BC:
x = (0 + 4) / 2 = 2
y = (6 - 3) / 2 = 1.5
So the midpoint of BC is N(2, 1.5).
Midpoint of CA:
x = (-3 + 4) / 2 = 0.5
y = (-3 + (-3)) / 2 = -3
So the midpoint of CA is P(0.5, -3).
Step 2: Draw lines from the midpoints to the opposite vertices.
Draw a line from M to C, from N to A, and from P to B.
Step 3: Find the point of intersection of the medians.
The point where the medians intersect is the centroid of the triangle.
To find the centroid, we need to find the intersection point of the lines MC, NA, and PB.
You can find the intersection point using various methods, such as solving the system of equations formed by the lines, using determinants, or using online tools like graphing calculators.
For this particular example, we will use the method of solving the system of equations.
Considering the equation of the line passing through points M and C:
y = mx + c,
Substituting values of M and C:
1.5 = m*(-1.5) + c ----(1)
Considering the equation of the line passing through points N and A:
y = mx + c,
Substituting values of N and A:
1.5 = m*(2) + c ----(2)
Considering the equation of the line passing through points P and B:
y = mx + c,
Substituting values of P and B:
-3 = m*(0.5) + c ----(3)
Now we have a system of three equations (equations 1, 2, and 3) with two unknowns (m and c). Solve this system of equations to find the values of m and c.
Once you find the values of m and c, substitute them back into any of the three equations to find the intersection point (x, y).
The intersection point will be the coordinates of the centroid of the triangle ABC.