a The Concurrency of Medians Theorem states the following conditional statement.
Concurrency of Medians Theorem
The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side.
Following the given steps, we will prove this theorem for the given triangle. We will begin by finding L, M, and N. To find these midpoints, we will use the Midpoint Formula.
(x_M,y_M) = (x_1+x_2/2, y_1+y_2/2 )
In this exercise, (x_1,y_1) and (x_2,y_2) are the endpoints of each side of the triangle. Let's start with L.
The coordinates of L are L( 1, 3). In the same way, we can calculate the coordinates of M and N.
End Points
Midpoint Formula
Midpoints Coordinates
B( 2, 6) & C( 8, 0)
(2+ 8/2, 6+ 0/2 )
M( 5, 3)
A( 0, 0) & C( 8, 0)
(0+ 8/2, 0+ 0/2 )
N( 4, 0)
b Now that we have the coordinates of the midpoints, we need to find the equations of AM, BN and CL. To find the equation of AM, first we need to find the slope of the line. To do that, we will use the Slope Formula.
m = y_2-y_1/x_2-x_1
In this case, (x_1,y_1) and (x_2,y_2) will be the vertices of the triangle and the opposite midpoints.
The equation of the line AM is y= 35x. Similarly, we can find the equations of the lines BN and CL.
Line
Points
m=y_2-y_1/x_2-x_1
y-y_1=m(x-x_1)
Equation
BN
B( 2, 6) & N( 4, 0)
0- 6/4- 2=-3
y- 6=-3(x- 2)
y=-3x+12
CL
C( 8, 0) & L( 1, 3)
3- 0/1- 8=-3/7
y- 0=-3/7(x- 8)
y=-3/7x+24/7
c Our third step is to find the coordinates of P, the centroid of the triangle. Since P is the intersection point of the lines AM and BN, to find its coordinates we need to solve the following system of equations.
y= 35x & (I) y=-3x+12 & (II)
Since the y-variable is already isolated (in both equations), we can solve the system using the Substitution Method.
Therefore, the coordinates of the centroid of â–ł ABC are P(103,2).
d Next, we need to check that P is on CL. To determine this, we need to substitute its coordinates into the equation of CL. If we get a true statement, then P is on CL. Otherwise, P is not on CL.
Since we got a true statement, we can conclude that P is on the line CL.
e Finally, we can verify the distance from each vertex to P.
Let's start with the median AM. We need to find AM and also AP. To do that, we will use the Distance Formula.
d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
Let's find AM!
From this, we can see that AP= 23AM. Using the Distance Formula again, we can find the remaining distances.
Endpoints
d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
Distance
B( 2, 6) & N( 4, 0)
BN=sqrt(( 4- 2)^2+( 0- 6)^2)
2sqrt(10)
B( 2, 6) & P(10/3,2)
BP=sqrt((10/3- 2)^2+(2- 6)^2)
4sqrt(10)/3 = 2/3* 2sqrt(10)
C( 8, 0) & L( 1, 3)
CL=sqrt(( 1- 8)^2+( 3- 0)^2)
sqrt(58)
C( 8, 0) & P(10/3,2)
CP=sqrt((10/3- 8)^2+(2- 0)^2)
2sqrt(58)/3
From the table, we have the following relations.
lc
BP = 4sqrt(10)/3 = 2/3BN & âś“
CP = 2sqrt(58)/3 = 2/3CL & âś“
Through this process, we have proven the Concurrency of Medians Theorem for â–ł ABC.
