To determine which set of numbers could represent the side lengths of a right triangle, we need to verify if the lengths satisfy the Pythagorean theorem. The Pythagorean theorem states that for a right triangle with side lengths [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] (where [tex]\(c\)[/tex] is the hypotenuse), the equation [tex]\(a^2 + b^2 = c^2\)[/tex] must hold true.
Here are the steps to verify each set:
1. Set: (15, 15, 21)
- Identify the longest side (hypotenuse): 21
- Check the Pythagorean theorem: [tex]\(15^2 + 15^2\)[/tex] should equal [tex]\(21^2\)[/tex]
- Calculation: [tex]\(15^2 = 225\)[/tex]; [tex]\(225 + 225 = 450\)[/tex]
- [tex]\(21^2 = 441\)[/tex]
- Since [tex]\(450 \neq 441\)[/tex], this set does not satisfy the theorem.
2. Set: (4, 8, 12)
- Identify the longest side (hypotenuse): 12
- Check the Pythagorean theorem: [tex]\(4^2 + 8^2\)[/tex] should equal [tex]\(12^2\)[/tex]
- Calculation: [tex]\(4^2 = 16\)[/tex]; [tex]\(8^2 = 64\)[/tex]; [tex]\(16 + 64 = 80\)[/tex]
- [tex]\(12^2 = 144\)[/tex]
- Since [tex]\(80 \neq 144\)[/tex], this set does not satisfy the theorem.
3. Set: (3, 5, 34)
- Identify the longest side (hypotenuse): 34
- Check the Pythagorean theorem: [tex]\(3^2 + 5^2\)[/tex] should equal [tex]\(34^2\)[/tex]
- Calculation: [tex]\(3^2 = 9\)[/tex]; [tex]\(5^2 = 25\)[/tex]; [tex]\(9 + 25 = 34\)[/tex]
- [tex]\(34^2 = 1156\)[/tex]
- Since [tex]\(34 \neq 1156\)[/tex], this set does not satisfy the theorem.
4. Set: (10, 24, 26)
- Identify the longest side (hypotenuse): 26
- Check the Pythagorean theorem: [tex]\(10^2 + 24^2\)[/tex] should equal [tex]\(26^2\)[/tex]
- Calculation: [tex]\(10^2 = 100\)[/tex]; [tex]\(24^2 = 576\)[/tex]; [tex]\(100 + 576 = 676\)[/tex]
- [tex]\(26^2 = 676\)[/tex]
- Since [tex]\(676 = 676\)[/tex], this set satisfies the theorem.
Therefore, out of all the options, the set of numbers [tex]\( (10, 24, 26) \)[/tex] could represent the side lengths, in inches, of a right triangle.
