Question

find the exact length of the missing side of the right triangle.

Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle with legs \(a\), \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). Here \(a = 3\), \(c=7\), and \(b=x\). So \(3^{2}+x^{2}=7^{2}\).

Step2: Simplify the equation

\(9 + x^{2}=49\). Then \(x^{2}=49 - 9\).

Step3: Calculate \(x^{2}\)

\(x^{2}=40\).

Step4: Solve for \(x\)

\(x=\sqrt{40}=\sqrt{4\times10}=2\sqrt{10}\) (since \(x>0\) as it represents the length of a side of a triangle).

Answer:

\(2\sqrt{10}\)