Question

question
the figure below is a square. find the length of side ( x ) in simplest radical form with a rational denominator.
(image of a square divided into two triangles by a diagonal, with one side labeled 3 and the diagonal labeled ( x ))

Explanation:

Step1: Analyze the square and triangle

The square is split into two congruent right - isosceles triangles by the diagonal. In a square, the diagonal divides it into two 45 - 45 - 90 triangles. The side length of the square (let's call it \(s = 3\)) and the diagonal \(x\) of the square are related by the Pythagorean theorem for a 45 - 45 - 90 triangle. For a 45 - 45 - 90 triangle, if the legs have length \(a\), the hypotenuse (diagonal in this case) \(c=a\sqrt{2}\). Here, the leg of the right - triangle (side of the square) \(a = 3\).

Step2: Apply the 45 - 45 - 90 triangle formula

We know that in a 45 - 45 - 90 triangle, the hypotenuse \(c=a\sqrt{2}\), where \(a\) is the length of each leg. Here, \(a = 3\) (the side length of the square), and the diagonal \(x\) is the hypotenuse of the right - triangle. So we substitute \(a = 3\) into the formula: \(x=3\sqrt{2}\)? Wait, no. Wait, actually, if we consider the square, when we draw a diagonal, the two triangles formed are right - angled isosceles triangles with legs equal to the side of the square and hypotenuse equal to the diagonal. Wait, maybe I mixed up. Wait, the side of the square is 3, and we want to find the diagonal \(x\). The Pythagorean theorem states that for a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(c^{2}=a^{2}+b^{2}\). In a square, \(a = b=3\), so \(x^{2}=3^{2}+3^{2}=9 + 9=18\). Then \(x=\sqrt{18}\). Simplify \(\sqrt{18}\): \(\sqrt{18}=\sqrt{9\times2}=3\sqrt{2}\). But wait, maybe the figure is such that the side of the square is related to the diagonal in a different way? Wait, no, the square has side length 3, and the diagonal is \(x\). So using Pythagoras, \(x=\sqrt{3^{2}+3^{2}}=\sqrt{18}=3\sqrt{2}\). Wait, but let's re - check. The square has side 3, so the two legs of the right - triangle (formed by the diagonal) are both 3. So \(x^{2}=3^{2}+3^{2}=18\), so \(x = \sqrt{18}=3\sqrt{2}\).

Answer:

\(3\sqrt{2}\)