Question

if a right triangle has legs of length 1 and 2, what is the length of the hypotenuse? if it has one leg of length 1 and a hypotenuse of the other leg? if a right triangle has legs of length 1 and x, what is the length of the hypotenuse?

Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(c^{2}=a^{2}+b^{2}\).

Step2: Solve for the hypotenuse when legs are 1 and 2

Let \(a = 1\) and \(b = 2\). Then \(c^{2}=1^{2}+2^{2}=1 + 4=5\), so \(c=\sqrt{5}\).

Step3: Solve for the other leg when one leg is 1 and hypotenuse is \(x\)

Let \(a = 1\) and \(c=x\). Then \(x^{2}=1^{2}+b^{2}\), so \(b^{2}=x^{2}-1\) and \(b=\sqrt{x^{2}-1}\) (assuming \(x>1\)).

Answer:

The length of the hypotenuse when legs are 1 and 2 is \(\sqrt{5}\). The length of the other leg when one leg is 1 and hypotenuse is \(x\) is \(\sqrt{x^{2}-1}\) (for \(x > 1\)).