Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Explore our Q&A platform to find in-depth answers from a wide range of experts in different fields. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
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].
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
