To determine if point $w$ is the midpoint of segment $xz$, we need to check if the distance from $x$ to $w$ is equal to the distance from $w$ to $z$.

Given:

• $xz = 10x + 50$
• $wz = 8x - 2$
• $wx = 4x - 2$

Step 1: Use the Segment Addition Postulate

According to the Segment Addition Postulate, the length of $xz$ should be equal to the sum of $wx$ and $wz$: $xz = wx + wz$

Step 2: Substitute the Given Values

Substituting the given expressions into the equation: $10x + 50 = (4x - 2) + (8x - 2)$

Step 3: Simplify the Right-Hand Side

Combine like terms on the right-hand side: $10x + 50 = 4x + 8x - 2 - 2$ $10x + 50 = 12x - 4$

Step 4: Solve for $x$

To solve for $x$, first get all the terms involving $x$ on one side of the equation: $10x - 12x = -4 - 50$ $-2x = -54$ $x = 27$

Step 5: Calculate the Lengths

Substitute $x = 27$ back into the expressions for $wx$ and $wz$: $wx = 4(27) - 2 = 108 - 2 = 106$ $wz = 8(27) - 2 = 216 - 2 = 214$

Step 6: Determine if $w$ is the Midpoint

For $w$ to be the midpoint, $wx$ should equal $wz$. But as we see: $wx = 106$ $wz = 214$ Since $wx \neq wz$, point $w$ is not the midpoint of $xz$.

Would you like more details on how to determine midpoints or how to solve similar problems?

Related Questions:

1. How do you find the midpoint of a segment given two endpoints?
2. What is the segment addition postulate, and how is it used in geometry?
3. Can you explain how to verify if a point is equidistant from two other points?
4. How would the solution change if the values for $wz$ or $wx$ were different?
5. What are the properties of collinear points in a line segment?

Tip:

In geometry, always remember that the midpoint of a segment divides the segment into two equal lengths. Use this property to check for midpoints in various problems.