Question

find the length of side ( x ) in simplest radical form with a rational denominator.
answer attempt 2 out of 2
( x = ) submit answer

Explanation:

Step1: Identify triangle type

The triangle is a right - isosceles triangle (two legs equal, right - angled), so by Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), here \(a = x\), \(b=x\), \(c = 1\).
$$x^{2}+x^{2}=1^{2}$$

Step2: Simplify the equation

Combine like terms: \(2x^{2}=1\). Then solve for \(x^{2}\): \(x^{2}=\frac{1}{2}\). Take square root: \(x=\sqrt{\frac{1}{2}}\) (we take positive root as length is positive).

Step3: Rationalize the denominator

Rationalize \(\sqrt{\frac{1}{2}}\) by multiplying numerator and denominator inside the square root by 2: \(x = \frac{\sqrt{2}}{2}\) (since \(\sqrt{\frac{1\times2}{2\times2}}=\frac{\sqrt{2}}{2}\)).

Answer:

\(\frac{\sqrt{2}}{2}\)