Question

pythagorean theorem (radical answers)
score: 1/5 penalty: 1 off
question
find the length of the third side. if necessary, write in simplest radical form.
(image of a right triangle with legs 3 and 3√3)
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, which states that for a right - triangle with legs of length \(a\) and \(b\) and hypotenuse of length \(c\), \(c^{2}=a^{2}+b^{2}\) (if we are finding the hypotenuse) or \(a^{2}=c^{2}-b^{2}\) (if we are finding a leg). Here, the two legs are \(a = 3\) and \(b=3\sqrt{3}\), and we need to find the hypotenuse \(c\).

Step2: Apply the Pythagorean theorem

Substitute \(a = 3\) and \(b = 3\sqrt{3}\) into the formula \(c^{2}=a^{2}+b^{2}\).
First, calculate \(a^{2}\): \(a^{2}=3^{2}=9\)
Then, calculate \(b^{2}\): \(b^{2}=(3\sqrt{3})^{2}=3^{2}\times(\sqrt{3})^{2}=9\times3 = 27\)
Now, find \(c^{2}\): \(c^{2}=9 + 27=36\)
Take the square root of both sides to find \(c\): \(c=\sqrt{36}=6\)

Answer:

\(6\)