Question

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

(7,24,25) (3,4,5)
(5,11,14) (6,8,10)
(6,10,12)

Explanation:

Step1: Recall Pythagorean theorem

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

Step2: Check \((7, 24, 25)\)

Calculate \(7^{2}+24^{2}\) and \(25^{2}\).
\(7^{2}=49\), \(24^{2}=576\), so \(7^{2}+24^{2}=49 + 576=625\).
\(25^{2}=625\). Since \(7^{2}+24^{2}=25^{2}\), \((7, 24, 25)\) is a Pythagorean triple.

Step3: Check \((3, 4, 5)\)

Calculate \(3^{2}+4^{2}\) and \(5^{2}\).
\(3^{2}=9\), \(4^{2}=16\), so \(3^{2}+4^{2}=9 + 16 = 25\).
\(5^{2}=25\). Since \(3^{2}+4^{2}=5^{2}\), \((3, 4, 5)\) is a Pythagorean triple.

Step4: Check \((5, 11, 14)\)

Calculate \(5^{2}+11^{2}\) and \(14^{2}\).
\(5^{2}=25\), \(11^{2}=121\), so \(5^{2}+11^{2}=25+121 = 146\).
\(14^{2}=196\). Since \(146
eq196\), \((5, 11, 14)\) is not a Pythagorean triple.

Step5: Check \((6, 8, 10)\)

Calculate \(6^{2}+8^{2}\) and \(10^{2}\).
\(6^{2}=36\), \(8^{2}=64\), so \(6^{2}+8^{2}=36 + 64=100\).
\(10^{2}=100\). Since \(6^{2}+8^{2}=10^{2}\), \((6, 8, 10)\) is a Pythagorean triple.

Step6: Check \((6, 10, 12)\)

Calculate \(6^{2}+10^{2}\) and \(12^{2}\).
\(6^{2}=36\), \(10^{2}=100\), so \(6^{2}+10^{2}=36+100 = 136\).
\(12^{2}=144\). Since \(136
eq144\), \((6, 10, 12)\) is not a Pythagorean triple.

Answer:

(7, 24, 25), (3, 4, 5), (6, 8, 10)