Given three collinear points X, Y, and Z and XY=8 and YZ=20, what is the length of a segment joining the midpoint XY and YZ?
Draw line segment XZ and let X = 0, the
starting point.
Xy = Y - 0 = 8,
Y = 8.
M1 = (8-0) / 2 = 4 = Mid-point of XY.
YZ = Z - Y = 20,
Z - 8 = 20,
Z = 28.
M2 - Y = 20 / 2 = 10,
M2 - 8 = 10,
M2 = 18.
Segment M1M2 = M2 - M1 = 18 - 4 = 14.
To find the length of the segment joining the midpoint of XY and YZ, we first need to find the coordinates of X, Y, and Z on a coordinate plane. Let's assume that X is located at (x1, y1), Y is located at (x2, y2), and Z is located at (x3, y3). Since the points are collinear, their x-coordinates or y-coordinates will have a common value.
Given that XY = 8 and YZ = 20, we can find the coordinates of Y by adding or subtracting the respective lengths from the coordinates of X and Z. Since XY = 8, we know that the coordinates of Y are the midpoint between the coordinates of X and Z.
Let's say X is located at (x1, y1) and Z is located at (x3, y3). The coordinates of Y can be calculated as follows:
x2 = (x1 + x3) / 2
y2 = (y1 + y3) / 2
Once we have the coordinates of Y, we can find the midpoint of XY and YZ. Let's call it point M. The coordinates of M can be calculated as follows:
xM = (x2 + x3) / 2
yM = (y2 + y3) / 2
Finally, we can use the distance formula to calculate the length of segment MY. The distance formula is given by:
d = √((x2 - xM)^2 + (y2 - yM)^2)
Substituting the values we calculated earlier, we can find the length of segment MY.