Get expert advice and community support for your questions on IDNLearn.com. Join our platform to receive prompt and accurate responses from experienced professionals in various fields.
Sagot :
To determine which of the given sets of side lengths will form a right triangle, we need to check if they satisfy the Pythagorean theorem. A right triangle will satisfy the equation [tex]\(a^2 + b^2 = c^2\)[/tex] for some permutation of the side lengths [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
Let's analyze each set individually:
### Set 1: [tex]\(10 \, \text{cm}, 26 \, \text{cm}, 24 \, \text{cm}\)[/tex]
1. Calculate [tex]\(10^2 + 26^2\)[/tex]:
[tex]\[ 10^2 + 26^2 = 100 + 676 = 776 \][/tex]
2. Calculate [tex]\(24^2\)[/tex]:
[tex]\[ 24^2 = 576 \][/tex]
Clearly, [tex]\(100 + 676 \neq 576\)[/tex].
3. Calculate [tex]\(10^2 + 24^2\)[/tex]:
[tex]\[ 10^2 + 24^2 = 100 + 576 = 676 \][/tex]
4. Calculate [tex]\(26^2\)[/tex]:
[tex]\[ 26^2 = 676 \][/tex]
Here, [tex]\(10^2 + 24^2 = 26^2\)[/tex], so Set 1 forms a right triangle.
### Set 2: [tex]\(4 \, \text{in}, 2 \, \text{in}, 6 \, \text{in}\)[/tex]
1. Calculate [tex]\(4^2 + 2^2\)[/tex]:
[tex]\[ 4^2 + 2^2 = 16 + 4 = 20 \][/tex]
2. Calculate [tex]\(6^2\)[/tex]:
[tex]\[ 6^2 = 36 \][/tex]
Clearly, [tex]\(16 + 4 \neq 36\)[/tex].
3. Calculate [tex]\(4^2 + 6^2\)[/tex]:
[tex]\[ 4^2 + 6^2 = 16 + 36 = 52 \][/tex]
4. Calculate [tex]\(2^2\)[/tex]:
[tex]\[ 2^2 = 4 \][/tex]
Clearly, [tex]\(16 + 36 \neq 4\)[/tex].
5. Calculate [tex]\(2^2 + 6^2\)[/tex]:
[tex]\[ 2^2 + 6^2 = 4 + 36 = 40 \][/tex]
6. Calculate [tex]\(4^2\)[/tex]:
[tex]\[ 4^2 = 16 \][/tex]
Clearly, [tex]\(4 + 36 \neq 16\)[/tex].
So, Set 2 does not form a right triangle.
### Set 3: [tex]\(14 \, \text{mm}, 13 \, \text{mm}, 15 \, \text{mm}\)[/tex]
1. Calculate [tex]\(14^2 + 13^2\)[/tex]:
[tex]\[ 14^2 + 13^2 = 196 + 169 = 365 \][/tex]
2. Calculate [tex]\(15^2\)[/tex]:
[tex]\[ 15^2 = 225 \][/tex]
Clearly, [tex]\(196 + 169 \neq 225\)[/tex].
3. Calculate [tex]\(14^2 + 15^2\)[/tex]:
[tex]\[ 14^2 + 15^2 = 196 + 225 = 421 \][/tex]
4. Calculate [tex]\(13^2\)[/tex]:
[tex]\[ 13^2 = 169 \][/tex]
Clearly, [tex]\(196 + 225 \neq 169\)[/tex].
5. Calculate [tex]\(13^2 + 15^2\)[/tex]:
[tex]\[ 13^2 + 15^2 = 169 + 225 = 394 \][/tex]
6. Calculate [tex]\(14^2\)[/tex]:
[tex]\[ 14^2 = 196 \][/tex]
Clearly, [tex]\(169 + 225 \neq 196\)[/tex].
So, Set 3 does not form a right triangle.
### Set 4: [tex]\(10 \, \text{ft}, 30 \, \text{ft}, 20 \, \text{ft}\)[/tex]
1. Calculate [tex]\(10^2 + 30^2\)[/tex]:
[tex]\[ 10^2 + 30^2 = 100 + 900 = 1000 \][/tex]
2. Calculate [tex]\(20^2\)[/tex]:
[tex]\[ 20^2 = 400 \][/tex]
Clearly, [tex]\(100 + 900 \neq 400\)[/tex].
3. Calculate [tex]\(10^2 + 20^2\)[/tex]:
[tex]\[ 10^2 + 20^2 = 100 + 400 = 500 \][/tex]
4. Calculate [tex]\(30^2\)[/tex]:
[tex]\[ 30^2 = 900 \][/tex]
Clearly, [tex]\(100 + 400 \neq 900\)[/tex].
5. Calculate [tex]\(20^2 + 30^2\)[/tex]:
[tex]\[ 20^2 + 30^2 = 400 + 900 = 1300 \][/tex]
6. Calculate [tex]\(10^2\)[/tex]:
[tex]\[ 10^2 = 100 \][/tex]
Clearly, [tex]\(400 + 900 \neq 100\)[/tex].
So, Set 4 does not form a right triangle.
Based on the calculations, the set that forms a right triangle is Set 1.
Let's analyze each set individually:
### Set 1: [tex]\(10 \, \text{cm}, 26 \, \text{cm}, 24 \, \text{cm}\)[/tex]
1. Calculate [tex]\(10^2 + 26^2\)[/tex]:
[tex]\[ 10^2 + 26^2 = 100 + 676 = 776 \][/tex]
2. Calculate [tex]\(24^2\)[/tex]:
[tex]\[ 24^2 = 576 \][/tex]
Clearly, [tex]\(100 + 676 \neq 576\)[/tex].
3. Calculate [tex]\(10^2 + 24^2\)[/tex]:
[tex]\[ 10^2 + 24^2 = 100 + 576 = 676 \][/tex]
4. Calculate [tex]\(26^2\)[/tex]:
[tex]\[ 26^2 = 676 \][/tex]
Here, [tex]\(10^2 + 24^2 = 26^2\)[/tex], so Set 1 forms a right triangle.
### Set 2: [tex]\(4 \, \text{in}, 2 \, \text{in}, 6 \, \text{in}\)[/tex]
1. Calculate [tex]\(4^2 + 2^2\)[/tex]:
[tex]\[ 4^2 + 2^2 = 16 + 4 = 20 \][/tex]
2. Calculate [tex]\(6^2\)[/tex]:
[tex]\[ 6^2 = 36 \][/tex]
Clearly, [tex]\(16 + 4 \neq 36\)[/tex].
3. Calculate [tex]\(4^2 + 6^2\)[/tex]:
[tex]\[ 4^2 + 6^2 = 16 + 36 = 52 \][/tex]
4. Calculate [tex]\(2^2\)[/tex]:
[tex]\[ 2^2 = 4 \][/tex]
Clearly, [tex]\(16 + 36 \neq 4\)[/tex].
5. Calculate [tex]\(2^2 + 6^2\)[/tex]:
[tex]\[ 2^2 + 6^2 = 4 + 36 = 40 \][/tex]
6. Calculate [tex]\(4^2\)[/tex]:
[tex]\[ 4^2 = 16 \][/tex]
Clearly, [tex]\(4 + 36 \neq 16\)[/tex].
So, Set 2 does not form a right triangle.
### Set 3: [tex]\(14 \, \text{mm}, 13 \, \text{mm}, 15 \, \text{mm}\)[/tex]
1. Calculate [tex]\(14^2 + 13^2\)[/tex]:
[tex]\[ 14^2 + 13^2 = 196 + 169 = 365 \][/tex]
2. Calculate [tex]\(15^2\)[/tex]:
[tex]\[ 15^2 = 225 \][/tex]
Clearly, [tex]\(196 + 169 \neq 225\)[/tex].
3. Calculate [tex]\(14^2 + 15^2\)[/tex]:
[tex]\[ 14^2 + 15^2 = 196 + 225 = 421 \][/tex]
4. Calculate [tex]\(13^2\)[/tex]:
[tex]\[ 13^2 = 169 \][/tex]
Clearly, [tex]\(196 + 225 \neq 169\)[/tex].
5. Calculate [tex]\(13^2 + 15^2\)[/tex]:
[tex]\[ 13^2 + 15^2 = 169 + 225 = 394 \][/tex]
6. Calculate [tex]\(14^2\)[/tex]:
[tex]\[ 14^2 = 196 \][/tex]
Clearly, [tex]\(169 + 225 \neq 196\)[/tex].
So, Set 3 does not form a right triangle.
### Set 4: [tex]\(10 \, \text{ft}, 30 \, \text{ft}, 20 \, \text{ft}\)[/tex]
1. Calculate [tex]\(10^2 + 30^2\)[/tex]:
[tex]\[ 10^2 + 30^2 = 100 + 900 = 1000 \][/tex]
2. Calculate [tex]\(20^2\)[/tex]:
[tex]\[ 20^2 = 400 \][/tex]
Clearly, [tex]\(100 + 900 \neq 400\)[/tex].
3. Calculate [tex]\(10^2 + 20^2\)[/tex]:
[tex]\[ 10^2 + 20^2 = 100 + 400 = 500 \][/tex]
4. Calculate [tex]\(30^2\)[/tex]:
[tex]\[ 30^2 = 900 \][/tex]
Clearly, [tex]\(100 + 400 \neq 900\)[/tex].
5. Calculate [tex]\(20^2 + 30^2\)[/tex]:
[tex]\[ 20^2 + 30^2 = 400 + 900 = 1300 \][/tex]
6. Calculate [tex]\(10^2\)[/tex]:
[tex]\[ 10^2 = 100 \][/tex]
Clearly, [tex]\(400 + 900 \neq 100\)[/tex].
So, Set 4 does not form a right triangle.
Based on the calculations, the set that forms a right triangle is Set 1.
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. For precise answers, trust IDNLearn.com. Thank you for visiting, and we look forward to helping you again soon.
