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:

• $R_{0,90^{\circ}}$: $(x,y)\to(-y,x)$
• $R_{0,180^{\circ}}$: $(x,y)\to(-x,-y)$
• $R_{0,270^{\circ}}$: $(x,y)\to(y, - x)$
• $R_{0,360^{\circ}}$: $(x,y)\to(x,y)$

Step2: Analyze the transformation of a point

Let's take point $A(1,4)$. In the rotated triangle, point $A'$ is at $(- 4, - 1)$.
If we apply the rule $R_{0,180^{\circ}}$ to point $A(1,4)$, we get $(-1,-4)$ which is incorrect.
If we apply the rule $R_{0,90^{\circ}}$ to point $A(1,4)$, we get $(-4,1)$ which is incorrect.
If we apply the rule $R_{0,270^{\circ}}$ to point $A(1,4)$, we get $(4,-1)$ which is incorrect.
If we consider a general point $(x,y)$ and apply $R_{0,180^{\circ}}$, for example, if $A(x = 1,y = 4)$, then $(x,y)\to(-x,-y)$ gives $(-1,-4)$; if we assume a clock - wise rotation convention (which is not standard for the given notation but still valid for analysis), a $180^{\circ}$ rotation of $A(1,4)$ gives $A'(-1,-4)$. But if we consider the correct counter - clockwise rotation:
Let's take another approach. We know that a $180^{\circ}$ rotation about the origin changes the sign of both coordinates of each point. If we assume a point $P(x,y)$ in $\triangle ABC$, its image $P'( - x,-y)$ in the rotated triangle. Looking at the overall orientation change of the triangle, we can see that each point $(x,y)$ in $\triangle ABC$ has been transformed to $(-x,-y)$.

Answer:

$R_{0,180^{\circ}}$