Question

question
which of the following sets of numbers could represent the three sides of a right triangle?

answer
{38, 80, 89} {65, 73, 97}
{40, 75, 85} {17, 63, 65}

Explanation:

Step1: Recall Pythagorean theorem

For a right triangle, the sum of the squares of the two shorter sides (legs) should equal the square of the longest side (hypotenuse), i.e., \(a^2 + b^2 = c^2\) where \(c\) is the hypotenuse.

Step2: Check set {38, 80, 89}

Calculate \(38^2 + 80^2\) and \(89^2\).
\(38^2 = 1444\), \(80^2 = 6400\), so \(38^2 + 80^2 = 1444 + 6400 = 7844\).
\(89^2 = 7921\). Since \(7844
eq 7921\), this is not a right triangle.

Step3: Check set {65, 73, 97}

Calculate \(65^2 + 73^2\) and \(97^2\).
\(65^2 = 4225\), \(73^2 = 5329\), so \(65^2 + 73^2 = 4225 + 5329 = 9554\).
\(97^2 = 9409\). Since \(9554
eq 9409\), this is not a right triangle.

Step4: Check set {40, 75, 85}

Calculate \(40^2 + 75^2\) and \(85^2\).
\(40^2 = 1600\), \(75^2 = 5625\), so \(40^2 + 75^2 = 1600 + 5625 = 7225\).
\(85^2 = 7225\). Since \(40^2 + 75^2 = 85^2\), this satisfies the Pythagorean theorem.

Step5: Check set {17, 63, 65} (optional, but for completeness)

Calculate \(17^2 + 63^2\) and \(65^2\).
\(17^2 = 289\), \(63^2 = 3969\), so \(17^2 + 63^2 = 289 + 3969 = 4258\).
\(65^2 = 4225\). Since \(4258
eq 4225\), this is not a right triangle.

Answer:

{40, 75, 85}