Question

1. which transformation carries the parallelogram below onto itself?

a) a rotation of 180° counterclockwise about the origin
b) a reflection over ( y = x )
c) a reflection over ( y = -x )
d) a rotation of 90° counterclockwise about the origin

Explanation:

Step1: Recall rotation rule

The rule for a \(180^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-x, -y)\).

Step2: Apply rotation rule to points

For point \(M(-2,3)\), after \(180^{\circ}\) rotation: \((-(-2),-3)=(2, - 3)\) (which is point \(T\)).
For point \(A(4,3)\), after \(180^{\circ}\) rotation: \((-4,-3)\) (which is point \(H\)).
For point \(H(-4,-3)\), after \(180^{\circ}\) rotation: \((4,3)\) (which is point \(A\)).
For point \(T(2,-3)\), after \(180^{\circ}\) rotation: \((-2,3)\) (which is point \(M\)).

Step3: Check other transformations

• For reflection over \(y = x\): The rule is \((x,y)\to(y,x)\). For \(M(-2,3)\), it becomes \((3,-2)\) which is not a vertex of the parallelogram.
• For reflection over \(y=-x\): The rule is \((x,y)\to(-y,-x)\). For \(M(-2,3)\), it becomes \((-3,2)\) which is not a vertex of the parallelogram.
• For \(90^{\circ}\) counter - clockwise rotation: The rule is \((x,y)\to(-y,x)\). For \(M(-2,3)\), it becomes \((-3,-2)\) which is not a vertex of the parallelogram.

Answer:

A. a rotation of \(180^{\circ}\) counterclockwise about the origin