Question

2-105. william thinks that the hypotenuse must be the longest side of a right triangle, but chad does not agree. who is correct? support your answer with an explanation or a counterexample, if possible. homework help

Explanation:

Step1: Recall Pythagorean theorem

In a right - triangle with sides \(a\), \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\), where \(a,b,c>0\).

Step2: Analyze side lengths

From \(a^{2}+b^{2}=c^{2}\), we know that \(c^{2}>a^{2}\) and \(c^{2}>b^{2}\) (since \(a,b > 0\)). Taking the square - root of both sides of \(c^{2}>a^{2}\) and \(c^{2}>b^{2}\) (and considering positive values for side lengths), we get \(c > a\) and \(c > b\). So the hypotenuse \(c\) is the longest side of a right - triangle.

Answer:

William is correct. The hypotenuse is the longest side of a right - triangle because of the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (\(a,b\) are the legs and \(c\) is the hypotenuse), which implies \(c>a\) and \(c > b\) for positive side lengths \(a,b,c\).