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Ella's geometry teacher asked each student to devise a problem and write out its solution. Here is Ella's work:

A triangle has side lengths of 10, 11, and 15. What type of triangle is it?

Procedure:
[tex]\[
\begin{array}{l}
10^2 \quad ? \quad 11^2 + 15^2 \\
100 \quad ? \quad 121 + 225 \\
100 \ \textless \ 346
\end{array}
\][/tex]

Conclusion:
This triangle is an acute triangle.

Which statement best summarizes Ella's work?

A. Ella's procedure and conclusion are correct.
B. Ella's procedure is correct, but her conclusion is incorrect.
C. Ella's procedure is incorrect, but her conclusion is correct.
D. Ella's procedure and conclusion are incorrect.



Answer :

GinnyAnswer
Let's analyze Ella's problem and solution step by step to determine if her procedure and conclusion are correct.

### Step-by-Step Analysis

1. Listing the side lengths of the triangle:
- Side 1: \(10\)
- Side 2: \(11\)
- Side 3: \(15\)

2. Calculating the squares of the sides:
- \(10^2 = 100\)
- \(11^2 = 121\)
- \(15^2 = 225\)

3. Checking if the triangle is acute:
- For a triangle to be acute, the sum of the squares of the two smaller sides must be greater than the square of the largest side.

Let's check the inequalities:
- \(100 < 121 + 225\)
[tex]\[ 100 < 346 \][/tex]
- \(121 < 100 + 225\)
[tex]\[ 121 < 325 \][/tex]
- \(225 < 100 + 121\)
[tex]\[ 225 < 221 \][/tex]

4. Evaluating the inequalities:
- \(100 < 346\) is true.
- \(121 < 325\) is true.
- \(225 < 221\) is false.

Since one of the inequalities \(225 < 221\) is false, the initial assumption that the triangle is acute is incorrect. An acute triangle must satisfy all three conditions where the sum of the squares of any two sides is greater than the square of the third side.

### Conclusion

- Ella's procedure of calculating the squares and comparing them is correct.
- However, her conclusion that the triangle is acute is incorrect because \(225 < 221\) is false.

Summary:
Ella's procedure is correct, but her conclusion is incorrect.

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