Answer :
To determine which set of side lengths could not represent the sides of a right triangle, we need to evaluate each set of measurements using the Pythagorean Theorem. According to the Pythagorean Theorem, for a triangle with side lengths [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] (where [tex]\( c \)[/tex] is the hypotenuse), the following relationship must hold true:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Let's evaluate each choice:
### Choice A: [tex]\( 3 \, \text{cm}, 4 \, \text{cm}, 5 \, \text{cm} \)[/tex]
We will check if [tex]\( 3^2 + 4^2 = 5^2 \)[/tex]:
[tex]\[ 3^2 = 9 \][/tex]
[tex]\[ 4^2 = 16 \][/tex]
[tex]\[ 9 + 16 = 25 \][/tex]
[tex]\[ 5^2 = 25 \][/tex]
Since [tex]\( 9 + 16 = 25 \)[/tex], the side lengths [tex]\( 3 \, \text{cm}, 4 \, \text{cm}, 5 \, \text{cm} \)[/tex] satisfy the Pythagorean Theorem. Thus, they can represent the side lengths of a right triangle.
### Choice B: [tex]\( 6 \, \text{cm}, 17.5 \, \text{cm}, 18.5 \, \text{cm} \)[/tex]
We will check if [tex]\( 6^2 + 17.5^2 = 18.5^2 \)[/tex]:
[tex]\[ 6^2 = 36 \][/tex]
[tex]\[ 17.5^2 = 306.25 \][/tex]
[tex]\[ 36 + 306.25 = 342.25 \][/tex]
[tex]\[ 18.5^2 = 342.25 \][/tex]
Since [tex]\( 36 + 306.25 = 342.25 \)[/tex], the side lengths [tex]\( 6 \, \text{cm}, 17.5 \, \text{cm}, 18.5 \, \text{cm} \)[/tex] satisfy the Pythagorean Theorem. Thus, they can represent the side lengths of a right triangle.
### Choice C: [tex]\( 5 \, \text{cm}, 12 \, \text{cm}, 13 \, \text{cm} \)[/tex]
We will check if [tex]\( 5^2 + 12^2 = 13^2 \)[/tex]:
[tex]\[ 5^2 = 25 \][/tex]
[tex]\[ 12^2 = 144 \][/tex]
[tex]\[ 25 + 144 = 169 \][/tex]
[tex]\[ 13^2 = 169 \][/tex]
Since [tex]\( 25 + 144 = 169 \)[/tex], these side lengths [tex]\( 5 \, \text{cm}, 12 \, \text{cm}, 13 \, \text{cm} \)[/tex] satisfy the Pythagorean Theorem. Thus, they can represent the side lengths of a right triangle.
### Choice D: [tex]\( 2 \, \text{cm}, 3 \, \text{cm}, 5 \, \text{cm} \)[/tex]
We will check if [tex]\( 2^2 + 3^2 = 5^2 \)[/tex]:
[tex]\[ 2^2 = 4 \][/tex]
[tex]\[ 3^2 = 9 \][/tex]
[tex]\[ 4 + 9 = 13 \][/tex]
[tex]\[ 5^2 = 25 \][/tex]
Since [tex]\( 4 + 9 = 13 \)[/tex] which is not equal to [tex]\( 25 \)[/tex], the side lengths [tex]\( 2 \, \text{cm}, 3 \, \text{cm}, 5 \, \text{cm} \)[/tex] do not satisfy the Pythagorean Theorem. Thus, they cannot represent the side lengths of a right triangle.
### Conclusion
The measurements that could not represent the side lengths of a right triangle are:
Choice D: [tex]\( 2 \, \text{cm}, 3 \, \text{cm}, 5 \, \text{cm} \)[/tex].
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Let's evaluate each choice:
### Choice A: [tex]\( 3 \, \text{cm}, 4 \, \text{cm}, 5 \, \text{cm} \)[/tex]
We will check if [tex]\( 3^2 + 4^2 = 5^2 \)[/tex]:
[tex]\[ 3^2 = 9 \][/tex]
[tex]\[ 4^2 = 16 \][/tex]
[tex]\[ 9 + 16 = 25 \][/tex]
[tex]\[ 5^2 = 25 \][/tex]
Since [tex]\( 9 + 16 = 25 \)[/tex], the side lengths [tex]\( 3 \, \text{cm}, 4 \, \text{cm}, 5 \, \text{cm} \)[/tex] satisfy the Pythagorean Theorem. Thus, they can represent the side lengths of a right triangle.
### Choice B: [tex]\( 6 \, \text{cm}, 17.5 \, \text{cm}, 18.5 \, \text{cm} \)[/tex]
We will check if [tex]\( 6^2 + 17.5^2 = 18.5^2 \)[/tex]:
[tex]\[ 6^2 = 36 \][/tex]
[tex]\[ 17.5^2 = 306.25 \][/tex]
[tex]\[ 36 + 306.25 = 342.25 \][/tex]
[tex]\[ 18.5^2 = 342.25 \][/tex]
Since [tex]\( 36 + 306.25 = 342.25 \)[/tex], the side lengths [tex]\( 6 \, \text{cm}, 17.5 \, \text{cm}, 18.5 \, \text{cm} \)[/tex] satisfy the Pythagorean Theorem. Thus, they can represent the side lengths of a right triangle.
### Choice C: [tex]\( 5 \, \text{cm}, 12 \, \text{cm}, 13 \, \text{cm} \)[/tex]
We will check if [tex]\( 5^2 + 12^2 = 13^2 \)[/tex]:
[tex]\[ 5^2 = 25 \][/tex]
[tex]\[ 12^2 = 144 \][/tex]
[tex]\[ 25 + 144 = 169 \][/tex]
[tex]\[ 13^2 = 169 \][/tex]
Since [tex]\( 25 + 144 = 169 \)[/tex], these side lengths [tex]\( 5 \, \text{cm}, 12 \, \text{cm}, 13 \, \text{cm} \)[/tex] satisfy the Pythagorean Theorem. Thus, they can represent the side lengths of a right triangle.
### Choice D: [tex]\( 2 \, \text{cm}, 3 \, \text{cm}, 5 \, \text{cm} \)[/tex]
We will check if [tex]\( 2^2 + 3^2 = 5^2 \)[/tex]:
[tex]\[ 2^2 = 4 \][/tex]
[tex]\[ 3^2 = 9 \][/tex]
[tex]\[ 4 + 9 = 13 \][/tex]
[tex]\[ 5^2 = 25 \][/tex]
Since [tex]\( 4 + 9 = 13 \)[/tex] which is not equal to [tex]\( 25 \)[/tex], the side lengths [tex]\( 2 \, \text{cm}, 3 \, \text{cm}, 5 \, \text{cm} \)[/tex] do not satisfy the Pythagorean Theorem. Thus, they cannot represent the side lengths of a right triangle.
### Conclusion
The measurements that could not represent the side lengths of a right triangle are:
Choice D: [tex]\( 2 \, \text{cm}, 3 \, \text{cm}, 5 \, \text{cm} \)[/tex].
