Question
Holly and Tamar completed the work in the table to determine if a triangle with side lengths of 40, 42, and 58 is a right triangle.
Holly’s Work
Tamar’s Work
(40 + 42) squared = 58 squared. 82 squared = 3,364. 6,724 not-equals 3,364.
The triangle is not a right triangle.
42 squared + 40 squared = 58 squared. 1,764 + 1,600 = 3,364. 3,364 = 3,364.
The triangle is a right triangle.
Which best describes the accuracy of their solutions?
Holly is correct.
Tamar is correct.
Neither Holly nor Tamar is correct.
Both Holly and Tamar are correct.
Holly’s Work
Tamar’s Work
(40 + 42) squared = 58 squared. 82 squared = 3,364. 6,724 not-equals 3,364.
The triangle is not a right triangle.
42 squared + 40 squared = 58 squared. 1,764 + 1,600 = 3,364. 3,364 = 3,364.
The triangle is a right triangle.
Which best describes the accuracy of their solutions?
Holly is correct.
Tamar is correct.
Neither Holly nor Tamar is correct.
Both Holly and Tamar are correct.
Answers
GPT-4o mini
To determine whether the triangle with side lengths of 40, 42, and 58 is a right triangle, we should apply the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides should equal the square of the longest side.
Let's break down both Holly's and Tamar's work step by step.
1. **Identifying the sides**: The side lengths are 40, 42, and 58. The longest side here is 58.
2. **Applying the Pythagorean theorem**: We check if \(40^2 + 42^2 = 58^2\).
- Calculating \(40^2\):
\[
40^2 = 1600
\]
- Calculating \(42^2\):
\[
42^2 = 1764
\]
- Adding these together:
\[
1600 + 1764 = 3364
\]
- Calculating \(58^2\):
\[
58^2 = 3364
\]
Since \(40^2 + 42^2 = 58^2\) (i.e., \(3364 = 3364\)), the conclusion is that the triangle is indeed a right triangle.
Now, reviewing Holly's and Tamar's work:
- Holly incorrectly stated the equation as \((40 + 42)^2 = 58^2\), which is not the correct application of the Pythagorean theorem. This leads her to conclude that the triangle is not a right triangle, which is incorrect.
- Tamar correctly applied the Pythagorean theorem with \(40^2 + 42^2 = 58^2\) and found the correct result, concluding that the triangle is a right triangle.
Thus, the correct answer is **Tamar is correct**.
Let's break down both Holly's and Tamar's work step by step.
1. **Identifying the sides**: The side lengths are 40, 42, and 58. The longest side here is 58.
2. **Applying the Pythagorean theorem**: We check if \(40^2 + 42^2 = 58^2\).
- Calculating \(40^2\):
\[
40^2 = 1600
\]
- Calculating \(42^2\):
\[
42^2 = 1764
\]
- Adding these together:
\[
1600 + 1764 = 3364
\]
- Calculating \(58^2\):
\[
58^2 = 3364
\]
Since \(40^2 + 42^2 = 58^2\) (i.e., \(3364 = 3364\)), the conclusion is that the triangle is indeed a right triangle.
Now, reviewing Holly's and Tamar's work:
- Holly incorrectly stated the equation as \((40 + 42)^2 = 58^2\), which is not the correct application of the Pythagorean theorem. This leads her to conclude that the triangle is not a right triangle, which is incorrect.
- Tamar correctly applied the Pythagorean theorem with \(40^2 + 42^2 = 58^2\) and found the correct result, concluding that the triangle is a right triangle.
Thus, the correct answer is **Tamar is correct**.