Let's carefully analyze Ella's work step-by-step to determine if her procedure and conclusion are correct:
1. Ella has a triangle with side lengths 10, 11, and 10.
2. She is trying to determine what type of triangle it is based on the lengths of its sides using some sort of inequality or property.
Let's correct a part of her work:
3. To determine the type of triangle based on side lengths, we should actually use the properties of triangles:
- If all sides are of equal length, it's an equilateral triangle.
- If exactly two sides are of equal length, it's an isosceles triangle.
- If none of the sides are of equal length, it's a scalene triangle.
- Additionally, we can use the Pythagorean Theorem to check for right triangles (a² + b² = c² for right triangles with c being the hypotenuse).
4. Ella's triangle sides are 10, 11, and 10. We notice that there are two sides of equal length (10 and 10), hence it is an isosceles triangle.
There seems to be a misunderstanding in Ella's work:
- Ella’s procedure involves squaring the sides and comparing them, suggesting she might be trying to use the Pythagorean theorem or properties related to the type of triangle based on angles, but her calculations are incorrect.
- Specifically:
- [tex]\(10^2 = 100\)[/tex]
- [tex]\(11^2 = 121\)[/tex]
- [tex]\(15^2 = 225\)[/tex] seems incorrect since 15 isn't one of the side lengths (should be squared).
- There may seem to be a misunderstanding of the comparison being made.
Conclusion:
- Ella's procedure is faulty as she has not used the correct method to determine the type of triangle.
- The correct determination of the triangle as isosceles and the acute nature of the triangle should be reevaluated if needed based on Pythagorean inequalities.
Therefore, the best statement summarizing Ella’s work is:
"O Ella's procedure is incorrect, but her conclusion is correct." (After properly adjusting for how type should be assessed based on side properties and confirming correct understanding of triangle type).
