Question

question
find the length of the third side. if necessary, write in simplest radical form.
(there is a right triangle with one leg labeled 7 and another leg labeled \\(\sqrt{51}\\), and the right angle is between these two legs. there is an input box for the answer and a submit answer button.)

Explanation:

Step1: Identify the triangle type

The triangle is a right - triangle, so we can use the Pythagorean theorem. Let the two legs be \(a = \sqrt{51}\) and \(b\) (the unknown side), and the hypotenuse be \(c = 7\). The Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\) (if \(c\) is the hypotenuse) or \(b^{2}+c^{2}=a^{2}\) (if \(a\) is the hypotenuse) or \(a^{2}+c^{2}=b^{2}\) (if \(b\) is the hypotenuse). We need to check which side is the hypotenuse. Since \(7=\sqrt{49}\) and \(\sqrt{51}>\sqrt{49}\), so \(\sqrt{51}\) is the hypotenuse. So the formula is \(a^{2}=b^{2}+c^{2}\), where \(a = \sqrt{51}\), \(c = 7\), and we solve for \(b\). Rearranging the formula, we get \(b^{2}=a^{2}-c^{2}\).

Step2: Substitute the values

Substitute \(a=\sqrt{51}\) and \(c = 7\) into the formula \(b^{2}=a^{2}-c^{2}\). We know that \(a^{2}=(\sqrt{51})^{2}=51\) and \(c^{2}=7^{2} = 49\). So \(b^{2}=51 - 49=2\).

Step3: Solve for \(b\)

Take the square root of both sides. Since \(b\) is a length, we take the positive square root. So \(b=\sqrt{2}\).

Answer:

\(\sqrt{2}\)