Let's use the Pythagorean theorem to determine whether each set of values forms a Pythagorean triple. Recall the Pythagorean theorem states that for a right triangle with sides [tex]\( a \)[/tex] and [tex]\( b \)[/tex] and hypotenuse [tex]\( c \)[/tex], it must be true that:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
We will check each set of values to see if this relationship holds.
1. Set (8, 15, 17)
[tex]\[
8^2 + 15^2 = 64 + 225 = 289
\][/tex]
[tex]\[
17^2 = 289
\][/tex]
Therefore, [tex]\( 8^2 + 15^2 = 17^2 \)[/tex], so (8, 15, 17) forms a Pythagorean triple. True
2. Set (1, √3, 2)
[tex]\[
1^2 + (\sqrt{3})^2 = 1 + 3 = 4
\][/tex]
[tex]\[
2^2 = 4
\][/tex]
Therefore, [tex]\( 1^2 + (\sqrt{3})^2 = 2^2 \)[/tex], so (1, √3, 2) forms a Pythagorean triple. False
3. Set (9, 12, 16)
[tex]\[
9^2 + 12^2 = 81 + 144 = 225
\][/tex]
[tex]\[
16^2 = 256
\][/tex]
Therefore, [tex]\( 9^2 + 12^2 \neq 16^2 \)[/tex], so (9, 12, 16) does not form a Pythagorean triple. False
4. Set (20, 21, 29)
[tex]\[
20^2 + 21^2 = 400 + 441 = 841
\][/tex]
[tex]\[
29^2 = 841
\][/tex]
Therefore, [tex]\( 20^2 + 21^2 = 29^2 \)[/tex], so (20, 21, 29) forms a Pythagorean triple. True
5. Set (8, 11, 14)
[tex]\[
8^2 + 11^2 = 64 + 121 = 185
\][/tex]
[tex]\[
14^2 = 196
\][/tex]
Therefore, [tex]\( 8^2 + 11^2 \neq 14^2 \)[/tex], so (8, 11, 14) does not form a Pythagorean triple. False
6. Set (30, 40, 50)
[tex]\[
30^2 + 40^2 = 900 + 1600 = 2500
\][/tex]
[tex]\[
50^2 = 2500
\][/tex]
Therefore, [tex]\( 30^2 + 40^2 = 50^2 \)[/tex], so (30, 40, 50) forms a Pythagorean triple. True
Hence, the results of verifying each set are:
1. (8, 15, 17): True
2. (1, √3, 2): False
3. (9, 12, 16): False
4. (20, 21, 29): True
5. (8, 11, 14): False
6. (30, 40, 50): True
