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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]\[10^2 \quad ? \quad 11^2 + 15^2\][/tex]
[tex]\[100 \quad ?? \quad 121 + 225\][/tex]
[tex]\[100 \ \textless \ 346\][/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 :

GinnyAnsweridn
Step-by-Step Solution:

1. Identify the side lengths of the triangle:
The triangle has side lengths of 10, 11, and 15.

2. Determine which side is the longest:
When dealing with triangles and checking for acute, right, or obtuse angles, it is often helpful to sort the side lengths. Here, the side lengths in ascending order are:
[tex]\(10 \leq 11 \leq 15\)[/tex]

3. Apply the Pythagorean Theorem to determine the type of triangle:
- For an acute triangle, the sum of the squares of the two shorter sides must be greater than the square of the longest side.
- For a right triangle, the sum of the squares of the two shorter sides must be equal to the square of the longest side.
- For an obtuse triangle, the sum of the squares of the two shorter sides must be less than the square of the longest side.

4. Square each side length:
- [tex]\(10^2 = 100\)[/tex]
- [tex]\(11^2 = 121\)[/tex]
- [tex]\(15^2 = 225\)[/tex]

5. Compare the sum of the squares of the two shorter sides to the square of the longest side:
- Sum of the squares of the two shorter sides: [tex]\(100 + 121 = 221\)[/tex]
- Square of the longest side: [tex]\(225\)[/tex]

6. Analyze the inequality:
- Since [tex]\(221\)[/tex] (sum of squares of the two shorter sides) is less than [tex]\(225\)[/tex] (square of the longest side), we determine that:
[tex]\(100 + 121 < 225\)[/tex]

7. Conclusion about the type of triangle:
Because the sum of the squares of the two shorter sides is less than the square of the longest side, the triangle is obtuse.

8. Evaluate Ella's procedure and conclusion:
- Ella intended to determine whether [tex]\(10^2 + 11^2 > 15^2\)[/tex]. However, based on the correct calculations, [tex]\(100 + 121 < 225\)[/tex], which shows that the triangle is obtuse.
- Therefore, Ella's conclusion that the triangle is acute is incorrect.

9. Summary of Ella's work:
- Procedure: Correctly carried out the initial steps to check if the triangle is acute, right, or obtuse by comparing squares of the sides.
- Conclusion: Incorrect, as the correct conclusion should have been that the triangle is obtuse.

The correct statement to summarize Ella's work is:
- Ella's procedure is correct, but her conclusion is incorrect.
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