To determine which of the given sets of numbers could represent the three sides of a right triangle, we need to use the Pythagorean theorem. The 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. Formally, for a right triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] (where [tex]\(c\)[/tex] is the hypotenuse):
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Let's analyze each set of numbers one by one:
### Set {49, 55, 73}
1. Identify the hypotenuse: The largest number is 73.
2. Check if the Pythagorean theorem holds:
[tex]\[
49^2 + 55^2 = 2401 + 3025 = 5426
\][/tex]
[tex]\[
73^2 = 5329
\][/tex]
Since [tex]\(5426 \neq 5329\)[/tex], the set {49, 55, 73} does not form a right triangle.
### Set {59, 63, 87}
1. Identify the hypotenuse: The largest number is 87.
2. Check if the Pythagorean theorem holds:
[tex]\[
59^2 + 63^2 = 3481 + 3969 = 7450
\][/tex]
[tex]\[
87^2 = 7569
\][/tex]
Since [tex]\(7450 \neq 7569\)[/tex], the set {59, 63, 87} does not form a right triangle.
### Set {48, 63, 80}
1. Identify the hypotenuse: The largest number is 80.
2. Check if the Pythagorean theorem holds:
[tex]\[
48^2 + 63^2 = 2304 + 3969 = 6273
\][/tex]
[tex]\[
80^2 = 6400
\][/tex]
Since [tex]\(6273 \neq 6400\)[/tex], the set {48, 63, 80} does not form a right triangle.
### Set {12, 35, 37}
1. Identify the hypotenuse: The largest number is 37.
2. Check if the Pythagorean theorem holds:
[tex]\[
12^2 + 35^2 = 144 + 1225 = 1369
\][/tex]
[tex]\[
37^2 = 1369
\][/tex]
Since [tex]\(1369 = 1369\)[/tex], the set {12, 35, 37} does form a right triangle.
### Conclusion
Among the given sets, only the set {12, 35, 37} can represent the three sides of a right triangle.
