To determine the validity of Ariel's work, let’s carefully review the steps and check all possible right triangle configurations. Here are the side lengths provided: 9, 15, and 12.
### Calculation Steps:
1. Calculate the squares of each side:
- [tex]\( 9^2 = 81 \)[/tex]
- [tex]\( 15^2 = 225 \)[/tex]
- [tex]\( 12^2 = 144 \)[/tex]
2. Check the Pythagorean Theorem with all possible combinations:
#### First Combination
Check if [tex]\( 9^2 + 15^2 = 12^2 \)[/tex]:
[tex]\[ 81 + 225 = 306 \][/tex]
[tex]\[ 306 \neq 144 \][/tex]
Hence, [tex]\( 9^2 + 15^2 \neq 12^2 \)[/tex].
#### Second Combination
Check if [tex]\( 9^2 + 12^2 = 15^2 \)[/tex]:
[tex]\[ 81 + 144 = 225 \][/tex]
[tex]\[ 225 = 225 \][/tex]
Hence, [tex]\( 9^2 + 12^2 = 15^2 \)[/tex].
#### Third Combination
Check if [tex]\( 12^2 + 15^2 = 9^2 \)[/tex]:
[tex]\[ 144 + 225 = 369 \][/tex]
[tex]\[ 369 \neq 81 \][/tex]
Hence, [tex]\( 12^2 + 15^2 \neq 9^2 \)[/tex].
### Conclusion:
- Ariel's statement that [tex]\( 9^2 + 15^2 = 12^2 \)[/tex] is incorrect because [tex]\( 306 \neq 144 \)[/tex].
- The correct approach should use the combination where [tex]\( 9^2 + 12^2 = 15^2 \)[/tex], which holds true since [tex]\( 225 = 225 \)[/tex].
Based on this information, the correct answer is:
No, Ariel should have added [tex]\( 9^2 \)[/tex] and [tex]\( 12^2 \)[/tex] and compared that to [tex]\( 15^2 \)[/tex].
