Question

triangle xyz is rotated to create the image triangle xyz
which rules could describe the rotation? select two options
$r_{0,90^{circ}}$
$r_{0,180^{circ}}$
$r_{0,270^{circ}}$
$(x,y)\to(-y,x)$
$(x,y)\to(-x,-y)$

Explanation:

Step1: Analyze rotation rules

• For a rotation of $180^{\circ}$ about the origin $R_{0,180^{\circ}}$, the rule is $(x,y)\to(-x, -y)$.
• Let's check the coordinates. Suppose $X(-2,2)$, $Y(2,2)$, $Z(0,4)$. After rotation, $X'=(2, - 2)$, $Y'=(-2,-2)$, $Z'=(0,-4)$.
• If we use the rule $(x,y)\to(-x, -y)$:
• For point $X(-2,2)$: $(-(-2),-2)=(2,-2)$
• For point $Y(2,2)$: $(-2,-2)$
• For point $Z(0,4)$: $(0,-4)$
• For a rotation of $90^{\circ}$ about the origin $R_{0,90^{\circ}}$, the rule is $(x,y)\to(-y,x)$. For example, if we take $X(-2,2)$, $(-2,2)\to(-2,-2)$ (not correct as per the image).
• For a rotation of $270^{\circ}$ about the origin $R_{0,270^{\circ}}$, the rule is $(x,y)\to(y, - x)$. For example, if we take $X(-2,2)$, $(-2,2)\to(2,2)$ (not correct as per the image).

Answer:

$R_{0,180^{\circ}}$, $(x,y)\to(-x, -y)$