Answer :
Let's analyze the problem step-by-step to reach a final assessment of Ella's work.
### Step-by-Step Solution
1. Side Lengths of the Triangle:
The sides of the triangle are given as 10, 11, and 15.
2. Square of Each Side:
Calculating the square of each side:
- [tex]\( 10^2 = 100 \)[/tex]
- [tex]\( 11^2 = 121 \)[/tex]
- [tex]\( 15^2 = 225 \)[/tex]
3. Sum of the Squares of the Smaller Sides:
Adding the squares of the two smaller sides:
- [tex]\( 10^2 + 11^2 = 100 + 121 = 221 \)[/tex]
4. Comparison for Type of Triangle:
To determine the type of triangle:
- For an acute triangle, the sum of the squares of any two sides must be greater than the square of the third side.
- For a right triangle, the Pythagorean theorem must hold, i.e., the square of the largest side equals the sum of the squares of the other two sides.
- For an obtuse triangle, the square of the largest side must be greater than the sum of the squares of the other two sides.
We need to compare:
- [tex]\( 10^2 + 11^2 \)[/tex] with [tex]\( 15^2 \)[/tex]
- From our calculations: [tex]\( 221 \)[/tex] (sum of two smaller sides squared) is less than [tex]\( 225 \)[/tex] (square of the largest side).
Clearly, [tex]\( 221 < 225 \)[/tex]. This suggests that the triangle is obtuse since the square of the longest side is greater than the sum of the squares of the other two sides.
5. Check Ella's Procedure:
- Ella compares [tex]\( 10^2 \)[/tex] with [tex]\( 11^2 + 15^2 \)[/tex]:
- She performed the comparison: [tex]\( 100 < 346 \)[/tex]
- This procedure effectively compares the smallest side with the sum of the other two, not directly identifying the type of triangle using the correct relationship.
- Even though her conclusion on identifying [tex]\( 100 < 346 \)[/tex] is mathematically accurate, it's not the correct method for determining the type of triangle based on side square comparisons.
- Ella's conclusion that the triangle is acute based on incorrectly applied logic is wrong.
### Conclusion
After analyzing Ella's work:
- Procedure Assessment: Ella’s calculation [tex]\( 100 < 346 \)[/tex] is correct but irrelevant to the correct method for determining the type of triangle.
- Conclusion Assessment: Ella concludes the triangle is acute, which is incorrect. The triangle, in fact, is obtuse.
### Final Assessment
The best summary of Ella's work would be:
"Ella's procedure is correct, but her conclusion is incorrect."
### Step-by-Step Solution
1. Side Lengths of the Triangle:
The sides of the triangle are given as 10, 11, and 15.
2. Square of Each Side:
Calculating the square of each side:
- [tex]\( 10^2 = 100 \)[/tex]
- [tex]\( 11^2 = 121 \)[/tex]
- [tex]\( 15^2 = 225 \)[/tex]
3. Sum of the Squares of the Smaller Sides:
Adding the squares of the two smaller sides:
- [tex]\( 10^2 + 11^2 = 100 + 121 = 221 \)[/tex]
4. Comparison for Type of Triangle:
To determine the type of triangle:
- For an acute triangle, the sum of the squares of any two sides must be greater than the square of the third side.
- For a right triangle, the Pythagorean theorem must hold, i.e., the square of the largest side equals the sum of the squares of the other two sides.
- For an obtuse triangle, the square of the largest side must be greater than the sum of the squares of the other two sides.
We need to compare:
- [tex]\( 10^2 + 11^2 \)[/tex] with [tex]\( 15^2 \)[/tex]
- From our calculations: [tex]\( 221 \)[/tex] (sum of two smaller sides squared) is less than [tex]\( 225 \)[/tex] (square of the largest side).
Clearly, [tex]\( 221 < 225 \)[/tex]. This suggests that the triangle is obtuse since the square of the longest side is greater than the sum of the squares of the other two sides.
5. Check Ella's Procedure:
- Ella compares [tex]\( 10^2 \)[/tex] with [tex]\( 11^2 + 15^2 \)[/tex]:
- She performed the comparison: [tex]\( 100 < 346 \)[/tex]
- This procedure effectively compares the smallest side with the sum of the other two, not directly identifying the type of triangle using the correct relationship.
- Even though her conclusion on identifying [tex]\( 100 < 346 \)[/tex] is mathematically accurate, it's not the correct method for determining the type of triangle based on side square comparisons.
- Ella's conclusion that the triangle is acute based on incorrectly applied logic is wrong.
### Conclusion
After analyzing Ella's work:
- Procedure Assessment: Ella’s calculation [tex]\( 100 < 346 \)[/tex] is correct but irrelevant to the correct method for determining the type of triangle.
- Conclusion Assessment: Ella concludes the triangle is acute, which is incorrect. The triangle, in fact, is obtuse.
### Final Assessment
The best summary of Ella's work would be:
"Ella's procedure is correct, but her conclusion is incorrect."
