Question

determine the length of the third side of the right triangle. write the answer as a simplified radical.
?
2 cm
8 cm
the length of the third side is cm.

Explanation:

Step1: Apply the Pythagorean theorem

The Pythagorean theorem for a right - triangle is \(a^{2}+b^{2}=c^{2}\), where \(a = 8\) (one leg), \(b = 2\) (another leg), and \(c\) is the hypotenuse (the side we want to find). So, \(c=\sqrt{a^{2}+b^{2}}\).
Substitute \(a = 8\) and \(b = 2\) into the formula: \(c=\sqrt{8^{2}+2^{2}}\).

Step2: Calculate the squares

\(8^{2}=64\) and \(2^{2}=4\). Then the expression becomes \(c=\sqrt{64 + 4}\).

Step3: Add the numbers inside the square root

\(64+4=68\). So, \(c=\sqrt{68}\).

Step4: Simplify the radical

Factor \(68\): \(68 = 4\times17\). Then \(\sqrt{68}=\sqrt{4\times17}\). Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (\(a = 4\), \(b = 17\)), we get \(\sqrt{4}\cdot\sqrt{17}\). Since \(\sqrt{4}=2\), the simplified form is \(2\sqrt{17}\).

Answer:

\(2\sqrt{17}\)