Certainly! Let's determine whether each set of values forms a Pythagorean triple by using the Pythagorean theorem:
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse [tex]\( c \)[/tex] is equal to the sum of the squares of the lengths of the other two sides [tex]\( a \)[/tex] and [tex]\( b \)[/tex]. This can be written as:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Let's verify each set one-by-one:
### 1. [tex]\((8, 15, 17)\)[/tex]
[tex]\[ a = 8, \quad b = 15, \quad c = 17 \][/tex]
Check if [tex]\( 8^2 + 15^2 = 17^2 \)[/tex]:
[tex]\[ 8^2 = 64 \][/tex]
[tex]\[ 15^2 = 225 \][/tex]
[tex]\[ 64 + 225 = 289 \][/tex]
[tex]\[ 17^2 = 289 \][/tex]
Since [tex]\( 8^2 + 15^2 = 17^2 \)[/tex], this set is a Pythagorean triple.
### 2. [tex]\((1, \sqrt{3}, 2)\)[/tex]
[tex]\[ a = 1, \quad b = \sqrt{3}, \quad c = 2 \][/tex]
Check if [tex]\( 1^2 + (\sqrt{3})^2 = 2^2 \)[/tex]:
[tex]\[ 1^2 = 1 \][/tex]
[tex]\[ (\sqrt{3})^2 = 3 \][/tex]
[tex]\[ 1 + 3 = 4 \][/tex]
[tex]\[ 2^2 = 4 \][/tex]
Since [tex]\( 1^2 + (\sqrt{3})^2 \neq 2^2 \)[/tex], this set is not a Pythagorean triple.
### 3. [tex]\((9, 12, 16)\)[/tex]
[tex]\[ a = 9, \quad b = 12, \quad c = 16 \][/tex]
Check if [tex]\( 9^2 + 12^2 = 16^2 \)[/tex]:
[tex]\[ 9^2 = 81 \][/tex]
[tex]\[ 12^2 = 144 \][/tex]
[tex]\[ 81 + 144 = 225 \][/tex]
[tex]\[ 16^2 = 256 \][/tex]
Since [tex]\( 9^2 + 12^2 \neq 16^2 \)[/tex], this set is not a Pythagorean triple.
### 4. [tex]\((8, 11, 14)\)[/tex]
[tex]\[ a = 8, \quad b = 11, \quad c = 14 \][/tex]
Check if [tex]\( 8^2 + 11^2 = 14^2 \)[/tex]:
[tex]\[ 8^2 = 64 \][/tex]
[tex]\[ 11^2 = 121 \][/tex]
[tex]\[ 64 + 121 = 185 \][/tex]
[tex]\[ 14^2 = 196 \][/tex]
Since [tex]\( 8^2 + 11^2 \neq 14^2 \)[/tex], this set is not a Pythagorean triple.
### 5. [tex]\((20, 21, 29)\)[/tex]
[tex]\[ a = 20, \quad b = 21, \quad c = 29 \][/tex]
Check if [tex]\( 20^2 + 21^2 = 29^2 \)[/tex]:
[tex]\[ 20^2 = 400 \][/tex]
[tex]\[ 21^2 = 441 \][/tex]
[tex]\[ 400 + 441 = 841 \][/tex]
[tex]\[ 29^2 = 841 \][/tex]
Since [tex]\( 20^2 + 21^2 = 29^2 \)[/tex], this set is a Pythagorean triple.
### 6. [tex]\((30, 40, 50)\)[/tex]
[tex]\[ a = 30, \quad b = 40, \quad c = 50 \][/tex]
Check if [tex]\( 30^2 + 40^2 = 50^2 \)[/tex]:
[tex]\[ 30^2 = 900 \][/tex]
[tex]\[ 40^2 = 1600 \][/tex]
[tex]\[ 900 + 1600 = 2500 \][/tex]
[tex]\[ 50^2 = 2500 \][/tex]
Since [tex]\( 30^2 + 40^2 = 50^2 \)[/tex], this set is a Pythagorean triple.
### Summary:
1. [tex]\((8, 15, 17)\)[/tex] is a Pythagorean triple.
2. [tex]\((1, \sqrt{3}, 2)\)[/tex] is not a Pythagorean triple.
3. [tex]\((9, 12, 16)\)[/tex] is not a Pythagorean triple.
4. [tex]\((8, 11, 14)\)[/tex] is not a Pythagorean triple.
5. [tex]\((20, 21, 29)\)[/tex] is a Pythagorean triple.
6. [tex]\((30, 40, 50)\)[/tex] is a Pythagorean triple.
