Question

Find the length of side x in simplest radical form with a rational denominator.45°V545°X

Answer 1

Since we have two angles of 45° and a right angle, we can deduce the opposite and the adjacent side are the same.

Using the pythagoras theorem we have,

$\begin{gathered} a^2+b^2=c^2 \\ x^2+x^2=\sqrt[]{5}^2\text{ Adding like terms we have} \\ 2x^2=5\text{ Isolating x, we get.} \\ x^2=\frac{5}{2}\text{ finding the root we have} \\ x=\sqrt[]{\frac{5}{2}}=\frac{\sqrt[]{5}}{\sqrt[]{2}} \end{gathered}$

Then, we have to find the simplest radical form, with the rational denominator. Rationalizing we have.

$\frac{\sqrt[]{5}}{\sqrt[]{2}}=\frac{\sqrt[]{5}}{\sqrt[]{2}}\cdot\frac{\sqrt[]{2}}{\sqrt[]{2}}=\frac{\sqrt[]{5\cdot2}}{\sqrt[]{2\cdot2}}=\frac{\sqrt[]{10}}{\sqrt[]{4}}=\frac{\sqrt[]{10}}{2}$

The final answer is

$x=\frac{\sqrt[]{10}}{2}$