Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
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.
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.
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
