Question

writing a rule for a rotation. what rule describes the rotation about the origin? (x, y) →. how many degrees was the figure rotated about the origin? (y, -x) (-y, x) (-x, -y) (x, y)

Explanation:

Step1: Recall rotation rules

In a 90 - degree counter - clockwise rotation about the origin, the rule is $(x,y)\to(-y,x)$. In a 90 - degree clockwise rotation about the origin, the rule is $(x,y)\to(y, - x)$. In a 180 - degree rotation about the origin, the rule is $(x,y)\to(-x,-y)$. In a 360 - degree rotation about the origin, the rule is $(x,y)\to(x,y)$.

Step2: Analyze the figure

By observing the original triangle $\triangle ABC$ and the rotated triangle $\triangle A'B'C'$, we can see that if we take a point $(x,y)$ in the original figure and its corresponding point in the rotated figure, the transformation is a 180 - degree rotation about the origin. The rule for a 180 - degree rotation about the origin is $(x,y)\to(-x,-y)$.

Answer:

$(x,y)\to(-x,-y)$; 180