Question

find the length of the third side. if necessary, write in simplest radical form.
image of a right triangle with a right angle, one leg labeled 4, and the hypotenuse labeled 6
answer attempt 1 out of 2
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Explanation:

Step1: Identify the triangle type

This is a right - triangle, so we can use the Pythagorean theorem. The Pythagorean theorem states that for a right - triangle with hypotenuse \(c\) and legs \(a\) and \(b\), \(c^{2}=a^{2}+b^{2}\). Here, the hypotenuse \(c = 6\), one leg \(a = 4\), and we need to find the other leg \(b\). Rearranging the formula for \(b\), we get \(b=\sqrt{c^{2}-a^{2}}\).

Step2: Substitute the values

Substitute \(c = 6\) and \(a = 4\) into the formula: \(b=\sqrt{6^{2}-4^{2}}\)

Step3: Calculate the squares

First, calculate \(6^{2}=36\) and \(4^{2}=16\). Then the expression becomes \(b=\sqrt{36 - 16}\)

Step4: Subtract inside the square root

\(36-16 = 20\), so \(b=\sqrt{20}\)

Step5: Simplify the radical

We can simplify \(\sqrt{20}\) as \(\sqrt{4\times5}\). Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a = 4\), \(b = 5\)), we get \(\sqrt{4}\times\sqrt{5}=2\sqrt{5}\)

Answer:

\(2\sqrt{5}\)