Question

which of the following sets of numbers could represent the three sides of a right triangle?
answer
{21, 72, 75}
{7, 8, 10}
{8, 12, 15}
{14, 84, 85}

Explanation:

Step1: Recall Pythagorean theorem

For a right triangle, \(a^2 + b^2 = c^2\) (where \(c\) is the hypotenuse, the longest side).

Step2: Check \(\{21, 72, 75\}\)

Calculate \(21^2 + 72^2\) and \(75^2\).
\(21^2 = 441\), \(72^2 = 5184\), so \(21^2 + 72^2 = 441 + 5184 = 5625\).
\(75^2 = 5625\). So \(21^2 + 72^2 = 75^2\).

Step3: Check \(\{8, 12, 15\}\)

\(8^2 = 64\), \(12^2 = 144\), \(8^2 + 12^2 = 64 + 144 = 208\).
\(15^2 = 225\). \(208
eq 225\).

Step4: Check \(\{7, 8, 10\}\)

\(7^2 = 49\), \(8^2 = 64\), \(7^2 + 8^2 = 49 + 64 = 113\).
\(10^2 = 100\). \(113
eq 100\).

Step5: Check \(\{14, 84, 85\}\)

\(14^2 = 196\), \(84^2 = 7056\), \(14^2 + 84^2 = 196 + 7056 = 7252\).
\(85^2 = 7225\). \(7252
eq 7225\).

Answer:

\(\{21, 72, 75\}\)