Question

triangle jkl is the image of triangle jkl under a rotation about the origin followed by a reflection across the y-axis.
write the rules for the rotation and reflection.
rotation: (x, y) ↦ (\boxed{ }, \quad )
reflection: (x, y) ↦ (\quad , \quad )

Explanation:

Step1: Determine the rotation rule

First, assume the rotation is \(180^{\circ}\). The rule for a \(180^{\circ}\) rotation about the origin is \((x,y)\to(-x, -y)\).
Take a point from \(\triangle JKL\), say \(J(-8,-3)\). After \(180^{\circ}\) rotation, it becomes \((8,3)\).

Step2: Determine the reflection rule

The rule for reflection across the \(y -\)axis is \((x,y)\to(-x,y)\).
Take the point \((8,3)\) (after rotation), after reflection across the \(y -\)axis, it becomes \((- 8,3)\) which is not correct.
Now assume the rotation is \(90^{\circ}\) counter - clockwise. The rule for a \(90^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-y,x)\).
Take \(J(-8,-3)\), after \(90^{\circ}\) counter - clockwise rotation: \((x,y)=(-8,-3)\to(3, - 8)\)
Then for reflection across the \(y -\)axis: \((x,y)\to(-x,y)\), so \((3,-8)\to(-3,-8)\) which is wrong.
Now assume the rotation is \(90^{\circ}\) clockwise. The rule for a \(90^{\circ}\) clockwise rotation about the origin is \((x,y)\to(y,-x)\).
Take \(J(-8,-3)\), after \(90^{\circ}\) clockwise rotation: \((x,y)=(-8,-3)\to(-3,8)\)
Then for reflection across the \(y -\)axis: \((x,y)\to(-x,y)\), so \((-3,8)\to(3,8)\) (matches \(J'\))
Take \(L(-6,-6)\), after \(90^{\circ}\) clockwise rotation: \((x,y)=(-6,-6)\to(-6,6)\)
Then after reflection across the \(y -\)axis: \((-6,6)\to(6,6)\) (matches \(L'\))
Take \(K(-2,-7)\), after \(90^{\circ}\) clockwise rotation: \((x,y)=(-2,-7)\to(-7,2)\)
Then after reflection across the \(y -\)axis: \((-7,2)\to(7,2)\) (matches \(K'\))

Answer:

Rotation: \((x,y)\to(y, - x)\)
Reflection: \((x,y)\to(-x,y)\)