Find the value of n and YZ if Y is between X and Z.

XY=5n, YZ=2n, XZ=91

Given the problem, we know that \( Y \) is between \( X \) and \( Z \), and the distances are as follows:

\[
XY = 5n, \quad YZ = 2n, \quad XZ = 91
\]

Since \( Y \) is between \( X \) and \( Z \), the sum of \( XY \) and \( YZ \) should equal \( XZ \):

\[
XY + YZ = XZ
\]

Substituting the given values:

\[
5n + 2n = 91
\]

Simplify and solve for \( n \):

\[
7n = 91
\]

\[
n = \frac{91}{7} = 13
\]

Now that we have \( n = 13 \), we can find \( YZ \):

\[
YZ = 2n = 2 \times 13 = 26
\]

So, the value of \( n \) is \( 13 \), and the length of \( YZ \) is \( 26 \).

Would you like more details, or do you have any further questions?

Here are 5 related questions for further practice:
1. If \( XY = 5n + 3 \) and \( YZ = 2n + 4 \), and \( XZ = 110 \), what is the value of \( n \)?
2. What happens if \( YZ = 3n \) instead of \( 2n \), and \( XZ \) remains 91?
3. If \( XY = 7n \) and \( YZ = 4n \), how would the value of \( n \) change if \( XZ = 121 \)?
4. Can \( YZ \) ever be greater than \( XY \)? Why or why not, assuming \( Y \) is still between \( X \) and \( Z \)?
5. If \( XZ \) were given as \( 91 + 2n \), how would you find \( n \)?

**Tip:** Always remember to check the relationship between the segments and their sum when solving geometry problems involving points on a line.

Learn how to find the value of n and the length of YZ when Y is between X and Z. This geometry problem involves using the segment addition postulate to solve for n and YZ given XY = 5n, YZ = 2n, and XZ = 91. Suitable for high school students studying geometry.

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