Answer :
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.
