Donc, côté = diagonale / √2 = (10√2) / √2 = 10 cm
["Understanding Diagonals: Why ( \frac{\ ext{diagonale}}{\sqrt{2}} = \frac{10\sqrt{2}}{\sqrt{2}} = 10 \ ext{ cm} ?", "When calculating diagonal lengths in squares or rectangles, a common formula arises: finding the diagonal from side measurements using the Pythagorean theorem. This article explains simply why\n[\n\frac{\ ext{diagonale}}{\sqrt{2}} = \frac{10\sqrt{2}}{\sqrt{2}} = 10 \ ext{ cm}\n]\nis a reliable way to solve diagonal problems.", "---", "### What Is a Diagonal in a Square?", "In geometry, the diagonal of a square connects two opposite corners, splitting the shape into two right-angled triangles. If each side of the square measures ( s ), then by the Pythagorean theorem, the diagonal ( d ) satisfies:\n[\nd^2 = s^2 + s^2 = 2s^2\n]\nTaking the square root gives:\n[\nd = \sqrt{2} \cdot s\n]\nSo, diagonal length equals side length multiplied by ( \sqrt{2} ).", "---", "### Simplifying the Ratio: Diagonal Over ( \sqrt{2} )", "From the formula above, dividing the diagonal by ( \sqrt{2} ) yields:\n[\n\frac{d}{\sqrt{2}} = \frac{\sqrt{2} \cdot s}{\sqrt{2}} = s\n]\nThis proves that dividing the diagonal of a square by ( \sqrt{2} ) returns the original side length. Conversely, multiplying the side by ( \sqrt{2} ) gives the diagonal.", "---", "### Applying the Formula in Practice", "Let’s take a specific example: suppose the side length is ( 10\sqrt{2} ) cm. Using the diagonal formula:\n[\nd = \sqrt{2} \cdot (10\sqrt{2}) = \sqrt{2} \cdot 10\sqrt{2} = 10 \cdot (\sqrt{2} \cdot \sqrt{2}) = 10 \cdot 2 = 20 \ ext{ cm}\n]\nBut wait — if instead the diagonal length is given as ( 10\sqrt{2} ) cm, then to recover the side:\n[\ns = \frac{10\sqrt{2}}{\sqrt{2}} = 10 \ ext{ cm}\n]\nThis confirms the formula’s validity: dividing a diagonal by ( \sqrt{2} ) returns the full side.", "---", "### Why √2 Is Crucial in Diagonal Calculations", "The number ( \sqrt{2} ) arises naturally from the geometry of right isosceles triangles: when the two legs are equal, the hypotenuse is ( \sqrt{2} ) times the leg. Using this principle in diagonals ensures accurate measurements in architecture, design, and engineering — where precise diagonal dimensions are essential.", "---", "### Final Takeaway", "Using the identity\n[\n\frac{\ ext{diagonale}}{\sqrt{2}} = \frac{10\sqrt{2}}{\sqrt{2}} = 10 \ ext{ cm}\n]\nis not just symbolic math — it’s a direct application of the Pythagorean theorem simplified through rationalization. This principle helps solve diagonal measurements quickly and accurately, whether in squares, rectangles, or other geometric figures where symmetry leads to ( \sqrt{2} ) relationships.", "Bottom line: Understanding how diagonal and side lengths relate via ( \sqrt{2} ) empowers better spatial reasoning and precise calculations — key in both math and real-world problem-solving.", "---\n*Keywords: diagonale berechnen, diagonal quadrats", "---", "By mastering this formula, you simplify diagonal problems and appreciate the elegant geometry underpinning everyday measurements."]
