Question

find the length of side x in simplest radical form with a rational denominator.
answer attempt 1 out of 2
x =

Explanation:

Step1: Identify the triangle type

The triangle is a right - isosceles triangle (since two legs are marked as equal, and it has a right angle). Let the length of each leg be \(x\), and the hypotenuse \(c = 1\).
For a right - triangle, we use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). In an isosceles right - triangle, \(a = b=x\) and \(c = 1\). So the formula becomes \(x^{2}+x^{2}=1^{2}\).

Step2: Simplify the equation

Combine like terms: \(2x^{2}=1\). Then, solve for \(x^{2}\) by dividing both sides by 2: \(x^{2}=\frac{1}{2}\).

Step3: Solve for \(x\)

Take the square root of both sides: \(x=\sqrt{\frac{1}{2}}\). To rationalize the denominator, we multiply the numerator and denominator inside the square root by 2: \(x = \frac{\sqrt{1\times2}}{\sqrt{2\times2}}=\frac{\sqrt{2}}{2}\). Since length cannot be negative, we take the positive square root.

Answer:

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