Question

which side lengths form a right triangle?
choose all answers that apply:
a 3,6,√45
b 2.5,6,6.5
c 4,8,√80

Explanation:

Step1: Check option A

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). Let \(a = 3\), \(b=6\), \(c=\sqrt{45}\).
\(a^{2}+b^{2}=3^{2}+6^{2}=9 + 36=45\)
\(c^{2}=(\sqrt{45})^{2}=45\)
Since \(a^{2}+b^{2}=c^{2}\), option A is correct.

Step2: Check option B

Let \(a = 2.5\), \(b = 6\), \(c=6.5\)
\(a^{2}+b^{2}=2.5^{2}+6^{2}=6.25+36 = 42.25\)
\(c^{2}=6.5^{2}=42.25\)
Since \(a^{2}+b^{2}=c^{2}\), option B is correct.

Step3: Check option C

Let \(a = 4\), \(b = 8\), \(c=\sqrt{80}\)
\(a^{2}+b^{2}=4^{2}+8^{2}=16 + 64=80\)
\(c^{2}=(\sqrt{80})^{2}=80\)
Since \(a^{2}+b^{2}=c^{2}\), option C is correct.

Answer:

A. \(3,6,\sqrt{45}\), B. \(2.5,6,6.5\), C. \(4,8,\sqrt{80}\)