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.
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
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