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 |
| [$\begin{array}{l}(40+42)^2=58^2 \\ 82^2=3,364 \\ 6,724 \approx 3,364\end{array}$] | [$\begin{array}{l}42^2+40^2=58^2 \\ 1,764+1,600=3,364 \\ 3,364=3,364\end{array}$] |
| The triangle is not a right triangle. | The triangle is a right triangle. |
Answer (2)
The triangle with side lengths 40, 42, and 58 is a right triangle, confirmed by the Pythagorean theorem. Tamar's calculations correctly show that the sum of the squares of the two shorter sides equals the square of the longest side. Holly's approach was flawed, leading to a wrong conclusion.
;
Holly incorrectly added the two shorter sides before squaring, which is not the Pythagorean theorem.
Tamar correctly squared each side and checked if the sum of the squares of the two shorter sides equals the square of the longest side.
Tamar's calculations are accurate: 4 0 2 + 4 2 2 = 1600 + 1764 = 3364 , and 5 8 2 = 3364 .
Therefore, the triangle is a right triangle, and Tamar is correct: The triangle is a right triangle.
Explanation
Analyze the problem and recall the Pythagorean theorem. Let's analyze the work of Holly and Tamar to determine who correctly applied the Pythagorean theorem to check if a triangle with side lengths 40, 42, and 58 is a right triangle. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as a 2 + b 2 = c 2 , where c is the length of the hypotenuse.
Check Holly's work. Holly's work: Holly calculated ( 40 + 42 ) 2 = 5 8 2 . This is incorrect because she added the two shorter sides before squaring. The Pythagorean theorem requires squaring each side individually before adding the squares of the two shorter sides. Therefore, Holly's approach is flawed, and her conclusion is likely incorrect.
Check Tamar's work. Tamar's work: Tamar calculated 4 0 2 + 4 2 2 = 5 8 2 . This is the correct application of the Pythagorean theorem. Let's verify her calculations: 4 0 2 = 1600 4 2 2 = 1764 5 8 2 = 3364 Now, let's check if 1600 + 1764 = 3364 :
1600 + 1764 = 3364 Since 3364 = 3364 , Tamar's calculation is correct, and the triangle with side lengths 40, 42, and 58 is indeed a right triangle.
Conclusion. Based on the correct application of the Pythagorean theorem, Tamar is correct. The triangle with side lengths 40, 42, and 58 is a right triangle because 4 0 2 + 4 2 2 = 5 8 2 . Holly's method was incorrect, leading to a wrong conclusion.
Examples
The Pythagorean theorem is a fundamental concept in construction and navigation. For example, builders use it to ensure that the corners of a building are square, guaranteeing structural integrity. Navigators use it to calculate distances and directions, especially when dealing with right triangles formed by their paths. This theorem is not just an abstract mathematical concept but a practical tool used in various real-world applications to ensure accuracy and stability.
