The coordinates of ∆LMN are L(4,7), M(6, −1), and N(−2,3).Algebraically, solve the system of equations represented by the medians
Determine the equations of two of the medians
To find the equations of the medians, we need to first find the midpoints of each of the sides of ∆LMN.
Midpoint of LM:
x-coordinate: (4+6)/2 = 5
y-coordinate: (7−1)/2 = 3
Midpoint of LN:
x-coordinate: (4−2)/2 = 1
y-coordinate: (7+3)/2 = 5
Midpoint of MN:
x-coordinate: (6−2)/2 = 2
y-coordinate: (−1+3)/2 = 1
Next, we need to find the slopes of two of the medians. A median is a line that passes through a vertex of a triangle and the midpoint of the opposite side.
Let's start with the median that passes through vertex L and the midpoint of side MN:
Midpoint of MN: (2,1)
Vertex L: (4,7)
Slope of median through L:
(1) Slope = (y2 - y1) / (x2 - x1)
Slope = (1 - 7) / (2 - 4)
Slope = 3
Equation of median through L:
(2) y - 7 = 3(x - 4)
Simplifying (2):
y - 7 = 3x - 12
y = 3x - 5
Now let's find the slope of the median that passes through vertex M and the midpoint of side LN:
Midpoint of LN: (1,5)
Vertex M: (6,-1)
Slope of median through M:
Slope = (y2 - y1) / (x2 - x1)
Slope = (-1 - 5) / (6 - 1)
Slope = -1.2
Equation of median through M:
(3) y - (-1) = -1.2(x - 6)
Simplifying (3):
y + 1 = -1.2x + 7.2
y = -1.2x + 6.2
Therefore, the equations of two of the medians are y = 3x - 5 and y = -1.2x + 6.2.