Question
Which statement describes whether a right triangle can be formed using one side length from each of these squares?
3 squares have areas of 64 inches squared, 225 inches squared, and 289 inches squared.
Yes, a right triangle can be formed because the sum of the areas of the two smaller squares does not equal the area of the largest square.
Yes, a right triangle can be formed because the sum of the areas of the two smaller squares equals the area of the largest square.
No, a right triangle cannot be formed because the sum of the areas of the two smaller squares does not equal the area of the largest square.
No, a right triangle cannot be formed because the sum of the areas of the two smaller squares equals the area of the largest square.
3 squares have areas of 64 inches squared, 225 inches squared, and 289 inches squared.
Yes, a right triangle can be formed because the sum of the areas of the two smaller squares does not equal the area of the largest square.
Yes, a right triangle can be formed because the sum of the areas of the two smaller squares equals the area of the largest square.
No, a right triangle cannot be formed because the sum of the areas of the two smaller squares does not equal the area of the largest square.
No, a right triangle cannot be formed because the sum of the areas of the two smaller squares equals the area of the largest square.
Answers
GPT-4o mini
To determine whether a right triangle can be formed using the side lengths of the squares with areas of 64 inches squared, 225 inches squared, and 289 inches squared, we first need to find the side lengths of these squares.
1. **Calculate the side lengths** using the areas:
- For the square with area 64 in²:
\[
\text{Side length} = \sqrt{64} = 8 \text{ inches}
\]
- For the square with area 225 in²:
\[
\text{Side length} = \sqrt{225} = 15 \text{ inches}
\]
- For the square with area 289 in²:
\[
\text{Side length} = \sqrt{289} = 17 \text{ inches}
\]
2. **Check if it forms a right triangle** using the Pythagorean theorem, which states that for a right triangle with side lengths \(a\), \(b\), and hypotenuse \(c\):
\[
a^2 + b^2 = c^2
\]
Here, we assign the side lengths:
- \(a = 8\)
- \(b = 15\)
- \(c = 17\)
3. **Calculate \(a^2 + b^2\) and \(c^2\)**:
\[
a^2 + b^2 = 8^2 + 15^2 = 64 + 225 = 289
\]
\[
c^2 = 17^2 = 289
\]
Since \(a^2 + b^2 = c^2\), the side lengths can indeed form a right triangle.
Given this analysis, the correct statement would be:
**Yes, a right triangle can be formed because the sum of the areas of the two smaller squares equals the area of the largest square.**
1. **Calculate the side lengths** using the areas:
- For the square with area 64 in²:
\[
\text{Side length} = \sqrt{64} = 8 \text{ inches}
\]
- For the square with area 225 in²:
\[
\text{Side length} = \sqrt{225} = 15 \text{ inches}
\]
- For the square with area 289 in²:
\[
\text{Side length} = \sqrt{289} = 17 \text{ inches}
\]
2. **Check if it forms a right triangle** using the Pythagorean theorem, which states that for a right triangle with side lengths \(a\), \(b\), and hypotenuse \(c\):
\[
a^2 + b^2 = c^2
\]
Here, we assign the side lengths:
- \(a = 8\)
- \(b = 15\)
- \(c = 17\)
3. **Calculate \(a^2 + b^2\) and \(c^2\)**:
\[
a^2 + b^2 = 8^2 + 15^2 = 64 + 225 = 289
\]
\[
c^2 = 17^2 = 289
\]
Since \(a^2 + b^2 = c^2\), the side lengths can indeed form a right triangle.
Given this analysis, the correct statement would be:
**Yes, a right triangle can be formed because the sum of the areas of the two smaller squares equals the area of the largest square.**