Step-by-Step Solution:
1. Identify the side lengths of the triangle:
The triangle has side lengths of 10, 11, and 15.
2. Determine which side is the longest:
When dealing with triangles and checking for acute, right, or obtuse angles, it is often helpful to sort the side lengths. Here, the side lengths in ascending order are:
[tex]\(10 \leq 11 \leq 15\)[/tex]
3. Apply the Pythagorean Theorem to determine the type of triangle:
- For an acute triangle, the sum of the squares of the two shorter sides must be greater than the square of the longest side.
- For a right triangle, the sum of the squares of the two shorter sides must be equal to the square of the longest side.
- For an obtuse triangle, the sum of the squares of the two shorter sides must be less than the square of the longest side.
4. Square each side length:
- [tex]\(10^2 = 100\)[/tex]
- [tex]\(11^2 = 121\)[/tex]
- [tex]\(15^2 = 225\)[/tex]
5. Compare the sum of the squares of the two shorter sides to the square of the longest side:
- Sum of the squares of the two shorter sides: [tex]\(100 + 121 = 221\)[/tex]
- Square of the longest side: [tex]\(225\)[/tex]
6. Analyze the inequality:
- Since [tex]\(221\)[/tex] (sum of squares of the two shorter sides) is less than [tex]\(225\)[/tex] (square of the longest side), we determine that:
[tex]\(100 + 121 < 225\)[/tex]
7. Conclusion about the type of triangle:
Because the sum of the squares of the two shorter sides is less than the square of the longest side, the triangle is obtuse.
8. Evaluate Ella's procedure and conclusion:
- Ella intended to determine whether [tex]\(10^2 + 11^2 > 15^2\)[/tex]. However, based on the correct calculations, [tex]\(100 + 121 < 225\)[/tex], which shows that the triangle is obtuse.
- Therefore, Ella's conclusion that the triangle is acute is incorrect.
9. Summary of Ella's work:
- Procedure: Correctly carried out the initial steps to check if the triangle is acute, right, or obtuse by comparing squares of the sides.
- Conclusion: Incorrect, as the correct conclusion should have been that the triangle is obtuse.
The correct statement to summarize Ella's work is:
- Ella's procedure is correct, but her conclusion is incorrect.
