Point w,x are collinear point w is between x and z xz= 10x+50 wz =8x-2 and wx=4x-2 use the info above is w the midpoint of xz

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.

Explore how to determine if point W is the midpoint of segment XZ using the Segment Addition Postulate and solving linear equations. This geometry problem involves calculating distances and verifying geometric properties. Suitable for high school geometry students.

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