Question

For the right triangle shown, the lengths of two sides are given. Find the third side. Leave your answer in simplified, radical form.

Answer 1

Answer: $c=4\sqrt{2}$

Step-by-step explanation:

For this exercise you need to apply the Pythagorean Theorem. This is:

$h^2=l^2+m^2$

Where "h" is the hypotenuse of the Right triangle and "l" and "m" are the legs.

In this case, you can identify that:

$h=c\\\\l=a=4\\\\m=b=4$

Knowing these values, you can substitute them into  $h^2=l^2+m^2$:

$c^2=4^2+4^2$

Now you must solve for "c":

$c=\sqrt{4^2+4^2}$

Evaluating, you get:

$c=\sqrt{16+16}\\\\c=\sqrt{32}$

To simplify the result:

- Descompose 32 into its prime factors:

$32=2*2*2*2*2=2^5$

- By the Product of powers property, you know that:

$2^5=2^4*2$

- Make the substitution:

$c=\sqrt{2^4*2}$

- Finally, knowing that $\sqrt[n]{a^n}=a$, you get:

$c=2^2\sqrt{2}\\\\c=4\sqrt{2}$