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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 :

Let's analyze Ella's problem and her solution step-by-step.

### Step-by-Step Solution:

1. Identification of the Sides:
Ella has identified the lengths of the sides of the triangle as 10, 11, and 15.

2. Determining the Type of Triangle:
To determine the type of triangle, Ella uses the properties of triangles based on their angles:

- An acute triangle is one where all angles are less than 90 degrees.
- A right triangle has one right angle (90 degrees).
- An obtuse triangle has one angle greater than 90 degrees.

3. Using the Pythagorean Theorem for Comparison:
The Pythagorean theorem ([tex]\(a^2 + b^2 = c^2\)[/tex]) is generally used for right triangles, but its relations help us in classifying triangles:

- For acute triangles: [tex]\(a^2 + b^2 > c^2\)[/tex]
- For right triangles: [tex]\(a^2 + b^2 = c^2\)[/tex]
- For obtuse triangles: [tex]\(a^2 + b^2 < c^2\)[/tex]

4. Calculations:
- Ella squares each side length:
- [tex]\(10^2 = 100\)[/tex] (side 'a')
- [tex]\(11^2 = 121\)[/tex] (side 'b')
- [tex]\(15^2 = 225\)[/tex] (side 'c')

5. Summation of Squares:
She compares the square of the shortest side (100) with the sum of the squares of the other two sides (121 + 225):
- [tex]\(100\)[/tex]
- [tex]\(121 + 225 = 346\)[/tex]

6. Comparison:
She compares [tex]\(100\)[/tex] with [tex]\(346\)[/tex]:
- [tex]\(100 < 346\)[/tex]

7. Conclusion:
Based on the comparison, since [tex]\(100 < 346\)[/tex], Ella concludes that the given triangle is an acute triangle because the square of one side is less than the sum of the squares of the other two sides.

### Summary of Analysis:
- Ella's procedure of squaring the sides and comparing them is correct.
- Her conclusion that [tex]\(100 < 346\)[/tex] indicates an acute triangle is correct.

Hence, the statement that best summarizes Ella's work is:

Ella's procedure and conclusion are correct.