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]
