Question

1. a rectangle is graphed on the set of axes below. does a counterclockwise rotation of 90° about point (3, -5) carry the rectangle onto itself? explain your answer

Explanation:

Step1: Recall the rotation rule

The rule for a counter - clockwise rotation of \(90^{\circ}\) about the origin is \((x,y)\to(-y,x)\).

Step2: Apply the rotation rule to the point \((3, - 5)\)

If \(x = 3\) and \(y=-5\), then after rotation, the new \(x\) - coordinate is \(-y=-(-5) = 5\) and the new \(y\) - coordinate is \(x = 3\). So the image of the point \((3,-5)\) after a \(90^{\circ}\) counter - clockwise rotation about the origin is \((5,3)\).

Step3: Check if the rectangle is invariant under the rotation

For a rectangle to be carried onto itself by a \(90^{\circ}\) counter - clockwise rotation about a point, the point must be the center of the rectangle. If we assume the center of the rectangle is \((h,k)\), and we use the rotation formula \((x,y)\to(x - h,y - k)\) (translate the center to the origin), then \((x - h,y - k)\to-(y - k)+h,(x - h)+k\) (apply \(90^{\circ}\) rotation), and then translate back. But if the center of the rectangle is not the origin, and we rotate about the origin, a non - square rectangle (since a square is a special case of a rectangle) does not have \(90^{\circ}\) rotational symmetry about a non - center point. A rectangle (non - square) has \(180^{\circ}\) rotational symmetry about its center. Since the rotation is about the origin (not the center of the rectangle, as we can see from the position of the point \((3,-5)\) not being the center of the rectangle in the coordinate system shown), a non - square rectangle does not have \(90^{\circ}\) rotational symmetry about the origin.

Answer:

No, a counter - clockwise rotation of \(90^{\circ}\) about the point \((3,-5)\) does not carry the rectangle onto itself. A non - square rectangle has \(180^{\circ}\) rotational symmetry about its center, not \(90^{\circ}\) rotational symmetry about a non - center point. If it were a square (a special rectangle), it would have \(90^{\circ}\) rotational symmetry about its center, but from the coordinate - based visual (assuming standard rectangle proportions from the grid), it is a non - square rectangle and the rotation is not about its center.