In a $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ right triangle, the lengths of the sides opposite the $30^{\circ}$, $60^{\circ}$, and $90^{\circ}$ angles are in what ratio?

A $30$-$60$-$90$ triangle is half of an equilateral triangle. If the hypotenuse (opposite $90^{\circ}$) has length $2x$, then the shorter leg (opposite $30^{\circ}$) is exactly half of it, length $x$. Applying the Pythagorean theorem: $x^{2} + y^{2} = (2x)^{2} \Rightarrow y^{2} = 4x^{2} - x^{2} = 3x^{2} \Rightarrow y = \sqrt{3}\, x$. So the ratio of sides opposite $30^{\circ}$ : $60^{\circ}$ : $90^{\circ}$ is $x : \sqrt{3}\, x : 2x = 1 : \sqrt{3} : 2$. - Trap A ($1 : 1 : \sqrt{2}$) is the **isosceles right triangle** ($45$-$45$-$90$), the *other* special right triangle. - B and C don't satisfy any Pythagorean relation.