Question

find the length of the third side. if necessary, write in simplest radical form.
right triangle with one leg labeled ( 3sqrt{3} ), hypotenuse labeled ( 6 ), and a right - angle symbol
answer attempt 1 out of 2
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radical symbol

Explanation:

Step1: Identify the triangle type

This is a right - triangle, so we can use the Pythagorean theorem, which states that for a right - triangle with legs \(a\), \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). Let the two known sides be \(a = 3\sqrt{3}\) (one leg) and \(c = 6\) (hypotenuse), and we need to find the other leg \(b\).

Step2: Apply the Pythagorean theorem

From \(a^{2}+b^{2}=c^{2}\), we can re - arrange it to solve for \(b\): \(b=\sqrt{c^{2}-a^{2}}\)
Substitute \(a = 3\sqrt{3}\) and \(c = 6\) into the formula:
First, calculate \(a^{2}=(3\sqrt{3})^{2}=3^{2}\times(\sqrt{3})^{2}=9\times3 = 27\)
Then, calculate \(c^{2}=6^{2}=36\)
Now, \(b=\sqrt{36 - 27}=\sqrt{9}=3\)

Answer:

\(3\)