Question

which of the following could represent lengths of the sides of a right triangle? select all that apply. 5, 12, 13 √6, √11, 4 3, 7.2, 7.8 6, 8, 10 4.6, 5.6, 6.5

Explanation:

Step1: Check 5, 12, 13

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). Let \(a = 5\), \(b=12\), \(c = 13\).
\(5^{2}+12^{2}=25 + 144=169\) and \(13^{2}=169\). So \(5^{2}+12^{2}=13^{2}\).

Step2: Check \(\sqrt{6},\sqrt{11},4\)

Let \(a=\sqrt{6}\), \(b = \sqrt{11}\), \(c = 4\).
\(a^{2}+b^{2}=(\sqrt{6})^{2}+(\sqrt{11})^{2}=6 + 11=17\) and \(c^{2}=4^{2}=16\). So \(a^{2}+b^{2}
eq c^{2}\).

Step3: Check 3, 7.2, 7.8

Let \(a = 3\), \(b=7.2\), \(c = 7.8\).
\(a^{2}+b^{2}=3^{2}+7.2^{2}=9+51.84 = 60.84\) and \(c^{2}=7.8^{2}=60.84\). So \(3^{2}+7.2^{2}=7.8^{2}\).

Step4: Check 6, 8, 10

Let \(a = 6\), \(b=8\), \(c = 10\).
\(a^{2}+b^{2}=6^{2}+8^{2}=36 + 64=100\) and \(c^{2}=10^{2}=100\). So \(6^{2}+8^{2}=10^{2}\).

Step5: Check 4.6, 5.6, 6.5

Let \(a = 4.6\), \(b=5.6\), \(c = 6.5\).
\(a^{2}+b^{2}=4.6^{2}+5.6^{2}=21.16+31.36 = 52.52\) and \(c^{2}=6.5^{2}=42.25\). So \(a^{2}+b^{2}
eq c^{2}\).

Answer:

5, 12, 13; 3, 7.2, 7.8; 6, 8, 10