Question

which are the side lengths of a right triangle? choose three correct answers.

1. 180, and 181
2. 19, and sqrt{380}
3. 9, and sqrt{85}
4. 14, and sqrt{205}
5. 11, and sqrt{158}

Explanation:

Step1: Use the Pythagorean theorem

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

Case1: For the set \(19,180,181\)

\(19^{2}+180^{2}=361 + 32400=32761\), and \(181^{2}=32761\). So \(19^{2}+180^{2}=181^{2}\)

Case2: For the set \(3,19,\sqrt{380}\)

\(3^{2}+19^{2}=9 + 361=370\), and \((\sqrt{380})^{2}=380\). Since \(370
eq380\)

Case3: For the set \(2,9,\sqrt{85}\)

\(2^{2}+9^{2}=4 + 81=85\), and \((\sqrt{85})^{2}=85\). So \(2^{2}+9^{2}=(\sqrt{85})^{2}\)

Case4: For the set \(3,14,\sqrt{205}\)

\(3^{2}+14^{2}=9+196 = 205\), and \((\sqrt{205})^{2}=205\). So \(3^{2}+14^{2}=(\sqrt{205})^{2}\)

Case5: For the set \(6,11,\sqrt{158}\)

\(6^{2}+11^{2}=36 + 121=157\), and \((\sqrt{158})^{2}=158\). Since \(157
eq158\)

Answer:

19, 180, and 181; 2, 9, and \(\sqrt{85}\); 3, 14, and \(\sqrt{205}\)