Question

rotation 4 the shape is rotated clockwise 270 degrees around the origin. can you find the new points? can you write the rule? what did you notice? the new points are... the rule is (x,y) →... i noticed...

Explanation:

Step1: Recall rotation rule

For a clockwise rotation of \(270^\circ\) around the origin, the rule is \((x,y)\to(-y,x)\).

Step2: Apply rule to each point

• For \((2,1)\):
• \(x = 2\), \(y = 1\). Using the rule, new \(x=-1\), new \(y = 2\). So new point is \((-1,2)\).
• For \((6,3)\):
• \(x = 6\), \(y = 3\). New \(x=-3\), new \(y = 6\). New point: \((-3,6)\).
• For \((8,5)\):
• \(x = 8\), \(y = 5\). New \(x=-5\), new \(y = 8\). New point: \((-5,8)\).

Step3: State the rule and notice

The rule is \((x,y)\to(-y,x)\). Notice that the \(x\) and \(y\) coordinates are transformed by negating the \(y\) and swapping with \(x\) (or related to \(90^\circ\) counter - clockwise rotation which has the same rule).

Answer:

The new points are \((-1,2)\), \((-3,6)\), \((-5,8)\).
The rule is \((x,y)\to(-y,x)\).
I noticed that a \(270^\circ\) clockwise rotation around the origin is equivalent to a \(90^\circ\) counter - clockwise rotation, and the coordinate transformation follows \((x,y)\to(-y,x)\).