To determine which set of numbers cannot represent the sides of a right triangle, we need to check if each set of numbers satisfies the Pythagorean theorem: [tex]\(a^2 + b^2 = c^2\)[/tex], where [tex]\( c \)[/tex] is the hypotenuse (the longest side) and [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the other two sides.
Let's check each set of sides one by one:
1. Set: \{45, 60, 75\}
- Sort the sides to ensure [tex]\( c \)[/tex] is the hypotenuse: [tex]\(a = 45\)[/tex], [tex]\(b = 60\)[/tex], [tex]\(c = 75\)[/tex].
- Calculate [tex]\(a^2 + b^2\)[/tex]:
[tex]\[
45^2 + 60^2 = 2025 + 3600 = 5625
\][/tex]
- Calculate [tex]\(c^2\)[/tex]:
[tex]\[
75^2 = 5625
\][/tex]
- Since [tex]\(a^2 + b^2 = c^2\)[/tex], this set represents the sides of a right triangle.
2. Set: \{30, 40, 50\}
- Sort the sides to ensure [tex]\( c \)[/tex] is the hypotenuse: [tex]\(a = 30\)[/tex], [tex]\(b = 40\)[/tex], [tex]\(c = 50\)[/tex].
- Calculate [tex]\(a^2 + b^2\)[/tex]:
[tex]\[
30^2 + 40^2 = 900 + 1600 = 2500
\][/tex]
- Calculate [tex]\(c^2\)[/tex]:
[tex]\[
50^2 = 2500
\][/tex]
- Since [tex]\(a^2 + b^2 = c^2\)[/tex], this set represents the sides of a right triangle.
3. Set: \{40, 42, 58\}
- Sort the sides to ensure [tex]\( c \)[/tex] is the hypotenuse: [tex]\(a = 40\)[/tex], [tex]\(b = 42\)[/tex], [tex]\(c = 58\)[/tex].
- Calculate [tex]\(a^2 + b^2\)[/tex]:
[tex]\[
40^2 + 42^2 = 1600 + 1764 = 3364
\][/tex]
- Calculate [tex]\(c^2\)[/tex]:
[tex]\[
58^2 = 3364
\][/tex]
- Since [tex]\(a^2 + b^2 = c^2\)[/tex], this set represents the sides of a right triangle.
4. Set: \{36, 47, 60\}
- Sort the sides to ensure [tex]\( c \)[/tex] is the hypotenuse: [tex]\(a = 36\)[/tex], [tex]\(b = 47\)[/tex], [tex]\(c = 60\)[/tex].
- Calculate [tex]\(a^2 + b^2\)[/tex]:
[tex]\[
36^2 + 47^2 = 1296 + 2209 = 3505
\][/tex]
- Calculate [tex]\(c^2\)[/tex]:
[tex]\[
60^2 = 3600
\][/tex]
- Since [tex]\(a^2 + b^2 \neq c^2\)[/tex], this set does not represent the sides of a right triangle.
Therefore, the set [tex]\(\{36, 47, 60\}\)[/tex] cannot represent the sides of a right triangle. The correct answer is:
[tex]\(\{36, 47, 60\}\)[/tex]
