Answered

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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]\[
\begin{array}{l}
10^2 \ \textless \ 11^2 + 15^2 \\
100 \ \textless \ 121 + 225 \\
100 \ \textless \ 346
\end{array}
\][/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.

Sagot :

GinnyAnswer
Let's evaluate Ella's procedure and conclusion step-by-step.

### Triangles and Their Classifications

Triangles can be classified based on their angles:
- Acute Triangle: All three angles are less than 90 degrees.
- Right Triangle: One angle is exactly 90 degrees.
- Obtuse Triangle: One angle is greater than 90 degrees.

### Using the Sides to Determine the Type of Triangle

To determine the type of triangle using the lengths of its sides, we use the Pythagorean Inequality Theorem:
- For an acute triangle, the square of the largest side must be less than the sum of the squares of the other two sides:
[tex]\[ c^2 < a^2 + b^2 \][/tex]
- For a right triangle, the square of the largest side must equal the sum of the squares of the other two sides:
[tex]\[ c^2 = a^2 + b^2 \][/tex]
- For an obtuse triangle, the square of the largest side must be greater than the sum of the squares of the other two sides:
[tex]\[ c^2 > a^2 + b^2 \][/tex]

### Given Data

The side lengths of the triangle are 10, 11, and 15. Let's identify the largest side, which is 15.

### Checking Ella's Procedure

1. Step 1: Calculate the square of the largest side:
[tex]\[ 15^2 = 225 \][/tex]

2. Step 2: Calculate the sum of the squares of the other two sides:
[tex]\[ 10^2 + 11^2 = 100 + 121 = 221 \][/tex]

3. Step 3: Compare the square of the largest side with the sum of the squares of the other two sides:
[tex]\[ 225 ? 221 \][/tex]

### Verify Ella's Statement

Ella's procedure states:
[tex]\[ 100 < 346 \][/tex]

However, we observe based on the calculations above:
[tex]\[ 225 > 221 \][/tex]

This indicates that the triangle is not acute. In fact, it is an obtuse triangle because the square of the largest side is greater than the sum of the squares of the other two sides.

### Conclusion

- Ella's procedure to check the type of triangle is methodologically correct.
- However, her conclusion that the triangle is acute is incorrect, as the correct conclusion based on the calculations should be that the triangle is obtuse.

### Summary

The best summary of Ella's work is:
"Ella's procedure is correct, but her conclusion is incorrect."
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