Question

question
find the length of the third side. if necessary, write in simplest radical form.
triangle with legs 7 and (4sqrt{2}), right - angled
answer attempt 1 out of 2

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 legs of lengths \(a\) 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), where \(c\) is the hypotenuse (the side opposite the right angle) and \(a\) and \(b\) are the legs. In this triangle, the two legs are \(a = 7\) and \(b=4\sqrt{2}\), and we need to find the hypotenuse \(c\).

Step2: Apply the Pythagorean theorem

First, calculate \(a^{2}\) and \(b^{2}\):
\(a^{2}=7^{2}=49\)
\(b^{2}=(4\sqrt{2})^{2}=4^{2}\times(\sqrt{2})^{2}=16\times2 = 32\)

Then, find \(c^{2}=a^{2}+b^{2}\)
\(c^{2}=49 + 32=81\)

Step3: Find the length of \(c\)

Take the square root of both sides: \(c=\sqrt{81}=9\) (since length cannot be negative)

Answer:

9