Question
Trisha did the following steps to determine that the sides 5, 13, and 12 do not form a right triangle. Do you agree with Trisha?
a2 + b2 = c2
52 + 132 = 122
25 + 169 = 144
194 ≠ 144
No, it should be a2 - b2 = c2
Yes
No, the side lengths should not be squared.
No, b should be 12 and c should be 13
a2 + b2 = c2
52 + 132 = 122
25 + 169 = 144
194 ≠ 144
No, it should be a2 - b2 = c2
Yes
No, the side lengths should not be squared.
No, b should be 12 and c should be 13
Answers
GPT-4o mini
Let's evaluate Trisha's steps and reasoning.
Trisha is using the Pythagorean theorem, which states that for a right triangle, the square of the length of the longest side (the hypotenuse) is equal to the sum of the squares of the lengths of the other two sides. She used the formula \(a^2 + b^2 = c^2\).
In the case of the sides 5, 13, and 12:
- The longest side is 13, so it should be treated as the hypotenuse (c).
- The other two sides are 5 and 12.
Trisha's work should actually be as follows:
1. Identify the sides: \(a = 5\), \(b = 12\), and \(c = 13\).
2. Compute the squares:
- \(5^2 + 12^2 = 13^2\)
- \(25 + 144 = 169\)
- \(169 = 169\)
Both sides of the equation equal, meaning the sides do form a right triangle.
To address the options:
- **No, it should be \(a^2 - b^2 = c^2\)**: Incorrect, the Pythagorean theorem is \(a^2 + b^2 = c^2\).
- **Yes**: Incorrect, she did not correctly conclude that these sides do NOT form a right triangle.
- **No, the side lengths should not be squared**: Incorrect, the sides must be squared to apply the Pythagorean theorem.
- **No, b should be 12 and c should be 13**: This is actually correct reasoning, but it reflects a misunderstanding of what she concluded.
In conclusion, Trisha's final statement is incorrect; the sides 5, 12, and 13 do indeed form a right triangle. Therefore, I do not agree with Trisha based on her calculations.
Trisha is using the Pythagorean theorem, which states that for a right triangle, the square of the length of the longest side (the hypotenuse) is equal to the sum of the squares of the lengths of the other two sides. She used the formula \(a^2 + b^2 = c^2\).
In the case of the sides 5, 13, and 12:
- The longest side is 13, so it should be treated as the hypotenuse (c).
- The other two sides are 5 and 12.
Trisha's work should actually be as follows:
1. Identify the sides: \(a = 5\), \(b = 12\), and \(c = 13\).
2. Compute the squares:
- \(5^2 + 12^2 = 13^2\)
- \(25 + 144 = 169\)
- \(169 = 169\)
Both sides of the equation equal, meaning the sides do form a right triangle.
To address the options:
- **No, it should be \(a^2 - b^2 = c^2\)**: Incorrect, the Pythagorean theorem is \(a^2 + b^2 = c^2\).
- **Yes**: Incorrect, she did not correctly conclude that these sides do NOT form a right triangle.
- **No, the side lengths should not be squared**: Incorrect, the sides must be squared to apply the Pythagorean theorem.
- **No, b should be 12 and c should be 13**: This is actually correct reasoning, but it reflects a misunderstanding of what she concluded.
In conclusion, Trisha's final statement is incorrect; the sides 5, 12, and 13 do indeed form a right triangle. Therefore, I do not agree with Trisha based on her calculations.