Question

which transformations can be used to carry abcd onto itself? the point of rotation is (3,2). check all that apply. a. rotation of 180° b. translation two units up c. reflection across the line y = 2 d. rotation of 90°

Explanation:

Step1: Analyze rotation of \(180^{\circ}\)

A rotation of \(180^{\circ}\) about the point \((3,2)\) will map each point \((x,y)\) of the rectangle \(ABCD\) to a new point \((6 - x,4 - y)\). Since the rectangle is symmetric about its center \((3,2)\), this rotation will carry \(ABCD\) onto itself.

Step2: Analyze translation two units up

A translation two units up will move the rectangle from \(y = 1\) and \(y = 3\) (for the lower and upper sides) to \(y=3\) and \(y = 5\). The original rectangle has points at \(y = 1\) and \(y = 3\), so this translation does not map \(ABCD\) onto itself.

Step3: Analyze reflection across \(y = 2\)

The line \(y = 2\) is the horizontal line midway between \(y = 1\) and \(y = 3\) (the \(y\) - coordinates of the sides of the rectangle). Reflecting a point \((x,y)\) across the line \(y = 2\) gives the point \((x,4 - y)\). Since the rectangle is symmetric about \(y = 2\), this reflection will carry \(ABCD\) onto itself.

Step4: Analyze rotation of \(90^{\circ}\)

A rotation of \(90^{\circ}\) about the point \((3,2)\) will change the orientation of the rectangle. The original rectangle has sides parallel to the \(x\) and \(y\) axes. After a \(90^{\circ}\) rotation, the sides will no longer be parallel to the \(x\) and \(y\) axes (assuming the rectangle is not a square), so this rotation does not map \(ABCD\) onto itself.

Answer:

A. Rotation of \(180^{\circ}\), C. Reflection across the line \(y = 2\)