Question

Find the length of side x in the simplest radical form with a rational denominator.

Answer 1

Answer:

$\sqrt{2}$

Step-by-step explanation:

We can use either angle, but I'm going to use the one on the bottom. So, in order to find x, we need to use tangent. One side we know is the adjacent, and the side we don't know is the opposite, therefore we need tangent. Here's the equation:

$tan(45)=\frac{x}{\sqrt{2}}$

Obviously, we can't have a root in our denominator, so we need to get rid of it somehow. Here's how:

We multiply the denominator of the fraction by $\sqrt{2}$. $\sqrt{2}$ multiplied by itself is simply 2. Try it! We also want to multiply the numerator by $\sqrt{2}$, but there isn't really a number we can use with that, so we'll just add it to the side. The equation you have now is:

$tan(45)=\frac{x\sqrt{2} }{2}$

Let's try to work this out now. Since the denominator is 2, we have to multiply both sides by it to find x.

$\frac{x\sqrt{2} }{2*2} =x\sqrt{2}$

$tan(45)*2=2$

We can plug 2 in for the x in the numerator now:

$\frac{2\sqrt{2} }{2}$

2 and 2 cancel out, so you get 1 in both the numerator and denominator. That's how we get our answer of $x=\sqrt{2}$

Also, because this is a 45-45-90 triangle, you don't really have to do all that work. If it's a 45-45-90 triangle, both legs should be the same length. :)