The coordinates of ∆LMN are L (4,7), M(6, −1), and N(−2,3)
Determine the equations of two of the medians.
To find the equation of a median, we need to first find the midpoint of the corresponding side and then find the midpoint of the opposite side. The median is the line that passes through these two midpoints.
Let's find the equation of the median that passes through point L. To do this, we need to find the midpoints of sides MN and LM.
Midpoint of MN:
x-coordinate = (−2 + 6)/2 = 2
y-coordinate = (3 − 1)/2 = 1
Midpoint of LM:
x-coordinate = (4 + 6)/2 = 5
y-coordinate = (7 − 1)/2 = 3
Now we can find the equation of the median that passes through L by using the two midpoints:
Slope of the median:
m = (1 − 7)/(2 − 5) = 2
Point-slope form of the equation:
y − 7 = 2(x − 4)
Simplify:
y = 2x − 1
Therefore, the equation of the median that passes through point L is y = 2x − 1.
Now let's find the equation of the median that passes through point M. To do this, we need to find the midpoints of sides LN and LM.
Midpoint of LN:
x-coordinate = (−2 + 4)/2 = 1
y-coordinate = (3 + 7)/2 = 5
Midpoint of LM:
x-coordinate = (4 + 6)/2 = 5
y-coordinate = (7 − 1)/2 = 3
Now we can find the equation of the median that passes through M by using the two midpoints:
Slope of the median:
m = (5 − (−1))/(1 − 6) = −4/5
Point-slope form of the equation:
y − (−1) = −4/5(x − 6)
Simplify:
y = −4/5x + 19/5
Therefore, the equation of the median that passes through point M is y = −4/5x + 19/5.