Question

triangle abc was rotated about the origin. which rule describes the rotation?
$r_{0,90^{circ}}$
$r_{0,180^{circ}}$
$r_{0,270^{circ}}$
$r_{0,360^{circ}}$

Explanation:

Step1: Recall rotation rules

For a rotation about the origin \(R_{0,\theta}\), we can analyze the change in coordinates of the vertices of the triangle. A \(90^{\circ}\) counter - clockwise rotation about the origin \((x,y)\to(-y,x)\), a \(180^{\circ}\) rotation \((x,y)\to(-x,-y)\), a \(270^{\circ}\) counter - clockwise rotation \((x,y)\to(y, - x)\) and a \(360^{\circ}\) rotation \((x,y)\to(x,y)\).

Step2: Observe the transformation

By looking at the original triangle \(ABC\) and the rotated triangle \(A'B'C'\), we can see that the coordinates of the vertices of the original triangle have changed signs for both \(x\) and \(y\) values. For example, if a vertex of \(\triangle ABC\) was \((x,y)\), the corresponding vertex of \(\triangle A'B'C'\) is \((-x,-y)\). This is the rule for a \(180^{\circ}\) rotation about the origin.

Answer:

\(R_{0,180^{\circ}}\)