In the diagram below of triangle J, K, LJKL, MM is the midpoint of start overline, J, L, end overline
JL
and NN is the midpoint of start overline, K, L, end overline
KL
. If M, N, equals, minus, 3, plus, 2, xMN=−3+2x, and J, K, equals, 29, minus, 3, xJK=29−3x, what is the measure of start overline, M, N, end overline
MN
?
Since M is the midpoint of JL and N is the midpoint of KL, MN is the segment connecting the midpoints of two sides of a triangle.
By the midpoint formula, the x-coordinate of M is the average of the x-coordinates of J and L, and the y-coordinate of M is the average of the y-coordinates of J and L.
x-coordinate of M: (x_J + x_L)/2 = (29 - 3x + 0)/2 = 29/2 - 3x/2
y-coordinate of M: (y_J+ y_L)/2 = (0 + 2x)/2 = x
Therefore, the coordinates of point M are (29/2 - 3x/2, x).
By the same logic, the coordinates of point N are (29/2 - 3x/2, x).
The distance between two points in a coordinate plane is given by the distance formula:
d = sqrt((x2-x1)^2 + (y2-y1)^2)
Substitute the coordinates of M and N into the distance formula:
MN = sqrt((29/2 - 3x/2 - 29/2)^2 + (x-x)^2)
MN = sqrt((-3x/2)^2)
MN = sqrt(9x^2/4)
MN = 3|x/2|
Therefore, the measure of segment MN is 3|x/2|.