Answer :
Let's evaluate Ella's work step by step.
First, let's summarize Ella's procedure:
1. She compared the square of one side of the triangle to the sum of the squares of the other two sides.
2. Specifically, she checked if [tex]\(10^2\)[/tex] is greater than [tex]\(11^2 + 15^2\)[/tex].
Next, let's evaluate the correctness of her procedure:
- According to the properties of triangles, to determine if a triangle is acute, we need to check if the sum of the squares of any two sides is greater than the square of the third side for all combinations. In other words:
- [tex]\(a^2 + b^2 > c^2\)[/tex]
- [tex]\(b^2 + c^2 > a^2\)[/tex]
- [tex]\(c^2 + a^2 > b^2\)[/tex]
- Ella's approach only checked one condition: [tex]\(10^2\)[/tex] compared to [tex]\(11^2 + 15^2\)[/tex]. This does not suffice to determine if the triangle is acute.
Let's look at the correct evaluation:
1. Calculate square of all sides:
- [tex]\(10^2 = 100\)[/tex]
- [tex]\(11^2 = 121\)[/tex]
- [tex]\(15^2 = 225\)[/tex]
2. Check all combinations:
- [tex]\(10^2 + 11^2 = 100 + 121 = 221\)[/tex]
- [tex]\(10^2 + 15^2 = 100 + 225 = 325\)[/tex]
- [tex]\(11^2 + 15^2 = 121 + 225 = 346\)[/tex]
Now compare these sums with the remaining side's square:
- [tex]\(221 > 225\)[/tex] is False
- [tex]\(325 > 121\)[/tex] is True
- [tex]\(346 > 100\)[/tex] is True
Since [tex]\(221\)[/tex] is not greater than [tex]\(225\)[/tex], the triangle cannot be acute. For it to be acute, all the conditions must be true, but one of them fails.
Conclusion:
- Ella's procedure was incorrect because she only checked one condition instead of all combinations.
- Her conclusion that the triangle is acute is also incorrect because one of the necessary conditions fails.
Thus, the correct statement is:
Ella's procedure and conclusion are incorrect.
First, let's summarize Ella's procedure:
1. She compared the square of one side of the triangle to the sum of the squares of the other two sides.
2. Specifically, she checked if [tex]\(10^2\)[/tex] is greater than [tex]\(11^2 + 15^2\)[/tex].
Next, let's evaluate the correctness of her procedure:
- According to the properties of triangles, to determine if a triangle is acute, we need to check if the sum of the squares of any two sides is greater than the square of the third side for all combinations. In other words:
- [tex]\(a^2 + b^2 > c^2\)[/tex]
- [tex]\(b^2 + c^2 > a^2\)[/tex]
- [tex]\(c^2 + a^2 > b^2\)[/tex]
- Ella's approach only checked one condition: [tex]\(10^2\)[/tex] compared to [tex]\(11^2 + 15^2\)[/tex]. This does not suffice to determine if the triangle is acute.
Let's look at the correct evaluation:
1. Calculate square of all sides:
- [tex]\(10^2 = 100\)[/tex]
- [tex]\(11^2 = 121\)[/tex]
- [tex]\(15^2 = 225\)[/tex]
2. Check all combinations:
- [tex]\(10^2 + 11^2 = 100 + 121 = 221\)[/tex]
- [tex]\(10^2 + 15^2 = 100 + 225 = 325\)[/tex]
- [tex]\(11^2 + 15^2 = 121 + 225 = 346\)[/tex]
Now compare these sums with the remaining side's square:
- [tex]\(221 > 225\)[/tex] is False
- [tex]\(325 > 121\)[/tex] is True
- [tex]\(346 > 100\)[/tex] is True
Since [tex]\(221\)[/tex] is not greater than [tex]\(225\)[/tex], the triangle cannot be acute. For it to be acute, all the conditions must be true, but one of them fails.
Conclusion:
- Ella's procedure was incorrect because she only checked one condition instead of all combinations.
- Her conclusion that the triangle is acute is also incorrect because one of the necessary conditions fails.
Thus, the correct statement is:
Ella's procedure and conclusion are incorrect.
