In each row, use the given side length for triangle XYZ to find the other side lengths of triangle XYZ.

[tex]\[
\begin{tabular}{|c|c|c|}
\hline
XY & XZ & YZ \\
\hline
10 & 24 & 26 \\
\hline
10 & 36 & \\
\hline
10 & & 32.5 \\
\hline
7.5 & & \\
\hline
\end{tabular}
\][/tex]



Answer :

To solve for the missing side lengths of the triangle [tex]$\triangle XYZ$[/tex], we use the given side lengths in each row and apply the Pythagorean theorem when necessary. Here's the detailed step-by-step solution:

### First Row
Given:
- [tex]\( XY = 10 \)[/tex]
- [tex]\( XZ = 24 \)[/tex]
- [tex]\( YZ = 26 \)[/tex]

Here, all three side lengths are provided:
- [tex]\( XY = 10 \)[/tex]
- [tex]\( XZ = 24 \)[/tex]
- [tex]\( YZ = 26 \)[/tex]

### Second Row
Given:
- [tex]\( XY = 10 \)[/tex]
- [tex]\( XZ = 36 \)[/tex]
- Find [tex]\( YZ \)[/tex]

To find the missing side [tex]\( YZ \)[/tex], we use the Pythagorean theorem in the form:
[tex]\[ YZ = \sqrt{XZ^2 - XY^2} \][/tex]
Substitute the given values:
[tex]\[ YZ = \sqrt{36^2 - 10^2} = \sqrt{1296 - 100} = \sqrt{1196} \approx 34.583232931581165 \][/tex]

Hence, the side lengths are:
- [tex]\( XY = 10 \)[/tex]
- [tex]\( XZ = 36 \)[/tex]
- [tex]\( YZ \approx 34.583232931581165 \)[/tex]

### Third Row
Given:
- [tex]\( XY = 10 \)[/tex]
- [tex]\( YZ = 32.5 \)[/tex]
- Find [tex]\( XZ \)[/tex]

To find the missing side [tex]\( XZ \)[/tex], we use the Pythagorean theorem in the form:
[tex]\[ XZ = \sqrt{YZ^2 - XY^2} \][/tex]
Substitute the given values:
[tex]\[ XZ = \sqrt{32.5^2 - 10^2} = \sqrt{1056.25 - 100} = \sqrt{956.25} \approx 30.923292192132454 \][/tex]

Hence, the side lengths are:
- [tex]\( XY = 10 \)[/tex]
- [tex]\( XZ \approx 30.923292192132454 \)[/tex]
- [tex]\( YZ = 32.5 \)[/tex]

### Fourth Row
Given:
- [tex]\( XY = 7.5 \)[/tex]
- [tex]\( XZ \)[/tex] is not provided
- [tex]\( YZ \)[/tex] is not provided

Since we have only one side length and no additional lengths or relationships given, [tex]\( XZ \)[/tex] and [tex]\( YZ \)[/tex] cannot be determined with the provided information.

Hence, the side lengths are:
- [tex]\( XY = 7.5 \)[/tex]
- [tex]\( XZ = \text{undetermined} \)[/tex]
- [tex]\( YZ = \text{undetermined} \)[/tex]

### Summary

The side lengths for each row are:
1. [tex]\( XY = 10, XZ = 24, YZ = 26 \)[/tex]
2. [tex]\( XY = 10, XZ = 36, YZ \approx 34.583232931581165 \)[/tex]
3. [tex]\( XY = 10, XZ \approx 30.923292192132454, YZ = 32.5 \)[/tex]
4. [tex]\( XY = 7.5, XZ = \text{undetermined}, YZ = \text{undetermined} \)[/tex]