Question

which of the following are pythagorean triples? select all that apply.
show your work here

(8,9,11) \t\t\t\t(9,12,15)
(8,15,17) \t\t\t\t(5,12,13)
(6,8,10)

Explanation:

A Pythagorean triple \((a, b, c)\) (where \(c\) is the largest number) satisfies the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). We will check each triple one by one.

Step 1: Check \((8, 9, 11)\)

Here, \(a = 8\), \(b = 9\), \(c = 11\).
Calculate \(a^{2}+b^{2}\): \(8^{2}+9^{2}=64 + 81=145\)
Calculate \(c^{2}\): \(11^{2}=121\)
Since \(145
eq121\), \((8, 9, 11)\) is not a Pythagorean triple.

Step 2: Check \((9, 12, 15)\)

Here, \(a = 9\), \(b = 12\), \(c = 15\).
Calculate \(a^{2}+b^{2}\): \(9^{2}+12^{2}=81+144 = 225\)
Calculate \(c^{2}\): \(15^{2}=225\)
Since \(225 = 225\), \((9, 12, 15)\) is a Pythagorean triple.

Step 3: Check \((8, 15, 17)\)

Here, \(a = 8\), \(b = 15\), \(c = 17\).
Calculate \(a^{2}+b^{2}\): \(8^{2}+15^{2}=64 + 225=289\)
Calculate \(c^{2}\): \(17^{2}=289\)
Since \(289=289\), \((8, 15, 17)\) is a Pythagorean triple.

Step 4: Check \((5, 12, 13)\)

Here, \(a = 5\), \(b = 12\), \(c = 13\).
Calculate \(a^{2}+b^{2}\): \(5^{2}+12^{2}=25+144 = 169\)
Calculate \(c^{2}\): \(13^{2}=169\)
Since \(169 = 169\), \((5, 12, 13)\) is a Pythagorean triple.

Step 5: Check \((6, 8, 10)\)

Here, \(a = 6\), \(b = 8\), \(c = 10\).
Calculate \(a^{2}+b^{2}\): \(6^{2}+8^{2}=36 + 64=100\)
Calculate \(c^{2}\): \(10^{2}=100\)
Since \(100 = 100\), \((6, 8, 10)\) is a Pythagorean triple.

Answer:

\((9, 12, 15)\), \((8, 15, 17)\), \((5, 12, 13)\), \((6, 8, 10)\)