Answer :
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.
### 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.
