Question

triangle qrs is transformed as shown on the graph. which rule describes the transformation? ( r_{0,90^{circ}} ) ( r_{0,180^{circ}} ) ( r_{0,270^{circ}} ) ( r_{0,360^{circ}} )

Explanation:

Step1: Recall rotation rules

For a point \((x,y)\) rotated about the origin \(O\):

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

Step2: Assume a point from \(\triangle QRS\)

Let's take point \(S(4,1)\). After transformation, assume \(S'\) has coordinates \((- 1,-4)\)

Step3: Check rotation rule

For \(R_{0,270^{\circ}}\):
If \((x = 4,y = 1)\), then \((y,-x)=(1,-4)\) (incorrect).
For \(R_{0,180^{\circ}}\):
If \((x = 4,y = 1)\), then \((-x,-y)=(-4,-1)\) (incorrect).
For \(R_{0,90^{\circ}}\):
If \((x = 4,y = 1)\), then \((-y,x)=(-1,4)\) (incorrect).
Let's take another approach.
The general rule for rotation:
If we consider the direction of rotation. A \(180^{\circ}\) rotation about the origin changes the sign of both \(x\) and \(y\) coordinates of all points of the figure.

Answer:

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