To determine which of the given sets of numbers could not represent the three sides of a right triangle, we need to verify if each set satisfies the Pythagorean theorem, which states that for a set of three numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] to form the sides of a right triangle, the relation [tex]\(a^2 + b^2 = c^2\)[/tex] must hold. Here, [tex]\(c\)[/tex] is the hypotenuse, the longest side of the triangle, and [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the other two sides.
Let's analyze each set of numbers one by one:
1. Set {9, 40, 41}:
- The numbers are 9, 40, and 41.
- Arrange them so that [tex]\(a \leq b \leq c\)[/tex], hence [tex]\(a=9\)[/tex], [tex]\(b=40\)[/tex], and [tex]\(c=41\)[/tex].
- Check the Pythagorean theorem: [tex]\(9^2 + 40^2 = 81 + 1600 = 1681\)[/tex] and [tex]\(41^2 = 1681\)[/tex].
Since [tex]\(81 + 1600 = 1681\)[/tex], which equals [tex]\(41^2\)[/tex], the set {9, 40, 41} can represent the sides of a right triangle.
2. Set {9, 12, 15}:
- The numbers are 9, 12, and 15.
- Arrange them so that [tex]\(a \leq b \leq c\)[/tex], hence [tex]\(a=9\)[/tex], [tex]\(b=12\)[/tex], and [tex]\(c=15\)[/tex].
- Check the Pythagorean theorem: [tex]\(9^2 + 12^2 = 81 + 144 = 225\)[/tex] and [tex]\(15^2 = 225\)[/tex].
Since [tex]\(81 + 144 = 225\)[/tex], which equals [tex]\(15^2\)[/tex], the set {9, 12, 15} can represent the sides of a right triangle.
3. Set {28, 44, 53}:
- The numbers are 28, 44, and 53.
- Arrange them so that [tex]\(a \leq b \leq c\)[/tex], hence [tex]\(a=28\)[/tex], [tex]\(b=44\)[/tex], and [tex]\(c=53\)[/tex].
- Check the Pythagorean theorem: [tex]\(28^2 + 44^2 = 784 + 1936 = 2720\)[/tex] and [tex]\(53^2 = 2809\)[/tex].
Since [tex]\(784 + 1936 = 2720\)[/tex], which is not equal to [tex]\(53^2 = 2809\)[/tex], the set {28, 44, 53} cannot represent the sides of a right triangle.
4. Set {39, 52, 65}:
- The numbers are 39, 52, and 65.
- Arrange them so that [tex]\(a \leq b \leq c\)[/tex], hence [tex]\(a=39\)[/tex], [tex]\(b=52\)[/tex], and [tex]\(c=65\)[/tex].
- Check the Pythagorean theorem: [tex]\(39^2 + 52^2 = 1521 + 2704 = 4225\)[/tex] and [tex]\(65^2 = 4225\)[/tex].
Since [tex]\(1521 + 2704 = 4225\)[/tex], which equals [tex]\(65^2\)[/tex], the set {39, 52, 65} can represent the sides of a right triangle.
So, the set of numbers that cannot represent the three sides of a right triangle is:
{28, 44, 53}
