Given: Y is the midpoint of segment XY
Prove: XY = 1/2 XZ
To prove the given statement, we need to show that the length of segment XY is equal to half the length of segment XZ.
To do this, we will use the definition of a midpoint and the properties of segments. The midpoint of a segment divides the segment into two congruent parts.
Let's assume that segment XZ has a length of 'd'. Since Y is the midpoint of segment XY, we can say that segment XY is divided into two equal parts.
Now, let's denote the length of segment XY as 'x'. Since Y is the midpoint of segment XY, we can also say that segment YZ is equal in length to segment XY.
According to the definition of a midpoint, we know that segment YZ has the same length as segment XY, and thus, it also has a length of 'x'.
By adding the length of segment XY and segment YZ, we get:
XY + YZ = x + x
Simplifying the equation gives us:
XY + YZ = 2x
Now, let's consider segment XZ. By adding the length of segment XY and segment YZ, we get:
XY + YZ = d
Since segment YZ is equal in length to segment XY, we can substitute 'x' for both XY and YZ in the equation:
2x = d
To prove that XY = 1/2 XZ, we must show that x = 1/2d. We can do this by dividing both sides of the equation by 2:
2x/2 = d/2
x = d/2
Therefore, we have shown that XY = 1/2 XZ. QED
Y cannot be the midpoint of XY. So let's call it M.
Since XM ≅ MY, their lengths are equal
length of XY is the sum of those two lengths
Now substitute, and divide by 2