Question
What is the image point of $(0,5)$ after the transformation $r_{y=x} \circ R_{180^{\circ}}$ ?
Answer
So, the image point of \((0,5)\) after the transformation \(r_{y=x} \circ R_{180^\circ}\) is \(\boxed{(-5,0)}\).
Step 1 ：Rotate the point \((0,5)\) by \(180^\circ\). The rotation of a point \((x,y)\) by \(180^\circ\) is given by the transformation \((-x,-y)\). So, the image of the point \((0,5)\) under the rotation \(R_{180^\circ}\) is \((-0,-5) = (0,-5)\).
Step 2 ：Reflect the point \((0,-5)\) in the line \(y=x\). The reflection of a point \((x,y)\) in the line \(y=x\) is given by the transformation \((y,x)\). So, the image of the point \((0,-5)\) under the reflection \(r_{y=x}\) is \((-5,0)\).
Step 3 ：So, the image point of \((0,5)\) after the transformation \(r_{y=x} \circ R_{180^\circ}\) is \(\boxed{(-5,0)}\).
