Prove this using direct proof in paragraph form
Given:M is the midpoint of XY
Prove: XM=YM
To prove that XM is equal to YM, we will use direct proof. We are given that M is the midpoint of XY. This means that M divides XY into two equal parts. By definition, a midpoint of a line segment divides the segment into two segments of equal length.
Let's consider the line segment XY. Since M is the midpoint, we can say that MX + MY = XY. However, we need to prove that XM is equal to YM, so let's re-arrange this equation.
We want to show that XM = YM. This means that we need to show that MX is equal to MY. From the equation MX + MY = XY, we can subtract MY from both sides to isolate MX. This gives us MX = XY - MY.
Now, since we know that M is the midpoint, we can substitute XY with 2M, as the segment is divided into two equal parts. Therefore, MX = 2M - MY.
Since M is the midpoint, we can also say that MX = MY, as they are both equal to half of XY. Therefore, our equation becomes MY = 2M - MY.
By simplifying the equation, we can add MY to both sides, giving us 2MY = 2M. To further simplify, we can divide both sides by 2, resulting in MY = M.
Hence, we have successfully proven that XM = YM, as MX is equal to MY.
To prove that XM = YM using a direct proof, we need to show that XM and YM are equal in length by using the given information that M is the midpoint of XY.
Let's begin the proof:
Assume that M is the midpoint of XY. This means that M divides XY into two equal parts. Now, let's consider the line segments XM and YM.
Since M is the midpoint of XY, we can say that the distance from X to M is equal to the distance from M to Y, as they are both half the length of XY. Denoting the distance from X to M as x and the distance from M to Y as y, we have:
XM = x
YM = y
Furthermore, since M is the midpoint, we know that the sum of the lengths XM and MY will be equal to the length of XY.
Therefore, XM + YM = XY
Substituting x for XM and y for YM, we have:
x + y = XY
But we also know that x is equal to y (the distances from X to M and from M to Y are equal), so we can further simplify the equation to:
2x = XY
Finally, since M is the midpoint of XY, and XM + YM = XY, we can conclude that XM = YM.
Hence, we have proven that XM is equal to YM using a direct proof.