Which set of numbers could be the lengths of the sides of a right triangle?
The set of numbers that could be the lengths of the sides of a right triangle must satisfy the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Therefore, the set of numbers that could be the lengths of the sides of a right triangle must follow this condition.
For example, one possible set of numbers could be (3, 4, 5). If we apply the Pythagorean theorem, we have:
3^2 + 4^2 = 9 + 16 = 25
5^2 = 25
Since both sides of the equation are equal, this set of numbers can be the lengths of the sides of a right triangle.
Other examples of sets of numbers that could be the lengths of the sides of a right triangle include:
- (5, 12, 13)
- (8, 15, 17)
- (7, 24, 25)
- (9, 40, 41)
In general, any set of positive integers (a, b, c) that satisfies a^2 + b^2 = c^2 could be the lengths of the sides of a right triangle.
To determine which set of numbers could be the lengths of the sides of a right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
So, let's consider the possible sets of numbers for the side lengths of a right triangle:
1. 3, 4, 5: This is a Pythagorean triple, and it satisfies the theorem. (3^2 + 4^2 = 9 + 16 = 25 = 5^2)
2. 5, 12, 13: Another Pythagorean triple that fits the theorem. (5^2 + 12^2 = 25 + 144 = 169 = 13^2)
3. 6, 8, 10: This is also a Pythagorean triple and satisfies the theorem. (6^2 + 8^2 = 36 + 64 = 100 = 10^2)
4. 7, 10, 12: Yet another Pythagorean triple that satisfies the theorem. (7^2 + 10^2 = 49 + 100 = 169 = 13^2)
These are just a few examples of sets of numbers that could be the lengths of the sides of a right triangle, but there are many more possibilities.