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 by a diagonal, with one side labeled \\(\sqrt{8}\\) and the diagonal labeled \\(x\\)
answer attempt 1 out of 2
\\(x = \\) blank submit answer

Explanation:

Step1: Recall square's diagonal formula

In a square, if side length is \( s \), diagonal \( d = s\sqrt{2} \). Here, side is \( \sqrt{8} \), so diagonal \( x \) (wait, no—wait, the triangle: the square's diagonal splits it into two right triangles. Wait, the side of the square is \( \sqrt{8} \), and \( x \) is the diagonal? Wait, no, the figure: the square has side \( \sqrt{8} \), and the diagonal \( x \)? Wait, no, let's correct. Wait, the square's side is \( \sqrt{8} \), and the diagonal \( x \) forms a right triangle with two sides of the square. So by Pythagorean theorem: \( x^2 = (\sqrt{8})^2 + (\sqrt{8})^2 \).

Step2: Calculate \( x^2 \)

\( (\sqrt{8})^2 = 8 \), so \( x^2 = 8 + 8 = 16 \)? Wait, no, that can't be. Wait, no—wait, maybe I mixed up. Wait, the square: each side is \( \sqrt{8} \), and the diagonal \( x \). Wait, no, Pythagorean theorem: in a square, diagonal \( d = s\sqrt{2} \), where \( s \) is side. So if side \( s = \sqrt{8} \), then diagonal \( x = s\sqrt{2} = \sqrt{8} \times \sqrt{2} \).

Step3: Simplify \( \sqrt{8} \times \sqrt{2} \)

\( \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \), so \( 2\sqrt{2} \times \sqrt{2} = 2 \times 2 = 4 \)? Wait, no, that's not right. Wait, no—wait, maybe the side is \( x \), and the diagonal is \( \sqrt{8} \)? Wait, the problem says "the figure below is a square. Find the length of side \( x \)". Wait, the diagram: the square has a diagonal? Wait, the triangle is a right triangle with legs \( x \) and \( x \) (since it's a square), and hypotenuse \( \sqrt{8} \)? Wait, that makes more sense. Oh! I had it reversed. So the square's side is \( x \), and the diagonal is \( \sqrt{8} \). So by Pythagorean theorem: \( x^2 + x^2 = (\sqrt{8})^2 \).

Step4: Solve for \( x \)

\( 2x^2 = 8 \) (since \( (\sqrt{8})^2 = 8 \)). Then \( x^2 = 4 \), so \( x = 2 \)? Wait, no, that's too simple. Wait, no—wait, maybe the diagonal is \( x \), and the side is \( \sqrt{8} \). Wait, let's re-express. Let's assume the square has side length \( s = \sqrt{8} \), and the diagonal is \( x \). Then by Pythagoras: \( x^2 = s^2 + s^2 = 2s^2 \). So \( x = s\sqrt{2} = \sqrt{8} \times \sqrt{2} \). Simplify \( \sqrt{8} \times \sqrt{2} = \sqrt{16} = 4 \). Wait, that's 4. But let's check again. Wait, \( \sqrt{8} = 2\sqrt{2} \), so \( 2\sqrt{2} \times \sqrt{2} = 2 \times 2 = 4 \). Yes. So if the side of the square is \( \sqrt{8} \), then the diagonal \( x \) is 4. But wait, the problem says "Find the length of side \( x \)". Wait, maybe the diagonal is \( \sqrt{8} \), and we need to find the side \( x \). Then: \( x\sqrt{2} = \sqrt{8} \), so \( x = \frac{\sqrt{8}}{\sqrt{2}} \). Rationalize denominator: \( \frac{\sqrt{8}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{16}}{2} = \frac{4}{2} = 2 \). Wait, now I'm confused. Let's look at the diagram again. The square has a diagonal drawn, splitting it into two triangles. One side of the square is labeled \( \sqrt{8} \), and the diagonal is labeled \( x \). Wait, no—wait, the square's sides are equal, so the two legs of the right triangle (the square's sides) are both \( \sqrt{8} \), and the hypotenuse (diagonal) is \( x \). Then by Pythagoras: \( x^2 = (\sqrt{8})^2 + (\sqrt{8})^2 = 8 + 8 = 16 \), so \( x = \sqrt{16} = 4 \). But if the side is \( x \), and the diagonal is \( \sqrt{8} \), then \( x\sqrt{2} = \sqrt{8} \), so \( x = \sqrt{8}/\sqrt{2} = \sqrt{4} = 2 \). But the problem says "the figure below is a square. Find the length of side \( x \)". Wait, the diagram: the square has a side labeled \( \sqrt{8} \), and the diagonal is \( x \)? Or the side is \( x \), and the diagonal is \( \sqrt{8} \)? Let's check the problem statement again: "Find the length of side \( x \)". So \( x \) is the side. Then the diagonal is \( \sqrt{8} \). So diagonal \( d = x\sqrt{2} \), so \( x\sqrt{2} = \sqrt{8} \). Solve for \( x \): \( x = \sqrt{8}/\sqrt{2} \). Simplify: \( \sqrt{8}/\sqrt{2} = \sqrt{8/2} = \sqrt{4} = 2 \). Wait, that's 2. But let's verify. If \( x = 2 \), then diagonal is \( 2\sqrt{2} \), but \( \sqrt{8} = 2\sqrt{2} \), so that's correct. So the side \( x \) is 2. Wait, but earlier when I thought the side was \( \sqrt{8} \), the diagonal was 4, but that would mean the side is \( \sqrt{8} \), diagonal 4, which is correct because \( \sqrt{8} \times \sqrt{2} = 4 \). But the problem says "Find the length of side \( x \)". So the side is \( x \), diagonal is \( \sqrt{8} \). So \( x\sqrt{2} = \sqrt{8} \), so \( x = \sqrt{8}/\sqrt{2} = \sqrt{4} = 2 \). Yes, that makes sense.

Answer:

\( x = 2 \)