On the unit circle, which of the following points would map onto itself after a reflection across the line $y=-x$?

• A. $(0,0)$
• B. $(1,0)$
• C. $(0,1)$
• D. $(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})$

Answer: (D)

Verified step by step guidance

1. Understand that reflecting a point across the line \(y = -x\) swaps the coordinates and changes their signs, specifically, a point \((x, y)\) maps to \((-y, -x)\) after reflection across this line.
2. To find points that map onto themselves after this reflection, set the original point equal to its image: \((x, y) = (-y, -x)\).
3. From the equality, derive the system of equations: \(x = -y\) and \(y = -x\).
4. Solve this system to find the relationship between \(x\) and \(y\). Both equations imply \(y = -x\), so points on the line \(y = -x\) are candidates for mapping onto themselves.
5. Since the points lie on the unit circle, use the equation \(x^2 + y^2 = 1\) together with \(y = -x\) to find the exact coordinates. Substitute \(y = -x\) into the circle equation to get \(x^2 + (-x)^2 = 1\), which simplifies to \$2x^2 = 1\(. Solve for \)x\( and then find \)y$.