Rectangle Side Ratio Problem: Finding Length m Using sqrt(x/2)

Break down $\sqrt{\frac{x}{2}}$ into $\frac{\sqrt{x}}{\sqrt{2}}$. This makes cross-multiplication easier: $\frac{x}{BC} = \frac{\sqrt{x}}{\sqrt{2}}$, so $BC = \frac{x\sqrt{2}}{\sqrt{x}}$.

The problem asks us to match one of the given answer choices. By adding 1 to both sides of $x^2 + 2x = m^2$, we get $(x+1)^2 = m^2 + 1$, which matches option D.

Yes! Since ABCD is a rectangle, triangle ABC is a right triangle. Use $AB^2 + BC^2 = AC^2$ where AC is the diagonal m.

Take it step by step. First find BC using the ratio, then substitute both AB and BC into the Pythagorean theorem. Simplify radicals early to avoid confusion later!

The problem specifically states that AC is labeled m. In a rectangle, both diagonals are equal, so it doesn't matter which one you calculate - they'll give the same answer.