Question

the figure below is a square. find the length of side ( x ) in simplest radical form with a rational denominator.

answer
( x = ) (sqrt{})

Explanation:

Step1: Identify the triangle type

The figure is a square, so the triangle formed by the diagonal and two sides is a right - isosceles triangle (since all angles in a square are \(90^{\circ}\) and the two sides of the square are equal). Let the side length of the square be \(a = 11\). We can use the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\), where in a square \(a = b=11\) and \(c=x\) (the diagonal).

Step2: Apply the Pythagorean theorem

Substitute \(a = 11\) and \(b = 11\) into the Pythagorean theorem:
\[

$$\begin{align*} x^{2}&=11^{2}+11^{2}\\ x^{2}&=121 + 121\\ x^{2}&=242 \end{align*}$$

\]

Step3: Simplify the radical

We know that \(242=121\times2\), so \(\sqrt{242}=\sqrt{121\times2}\). Using the property of square roots \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\)), we have \(\sqrt{121\times2}=\sqrt{121}\times\sqrt{2}=11\sqrt{2}\). Since \(x\) represents a length, \(x>0\), so \(x = 11\sqrt{2}\).

Answer:

\(11\sqrt{2}\)