Question

1. antoine and bobby each rotated a pentagon about point p, but they each got a different image. which rotation is correct? why? 2. a. draw rotations of rectangle stuv 90°, 180°, and 270° clockwise about the origin. b. what do all four figures have in common? they all have side lengths and angles measures, and their sides are parallel.

Explanation:

Step1: Recall rotation rules

For a rotation of a point $(x,y)$ about the origin:

• A $90^{\circ}$ clock - wise rotation gives $(y, - x)$.
• A $180^{\circ}$ clock - wise rotation gives $(-x,-y)$.
• A $270^{\circ}$ clock - wise rotation gives $(-y,x)$.

Step2: Analyze pentagon rotation

When rotating a polygon about a point, the distance of each vertex from the center of rotation (Point $P$ in this case) remains the same. The orientation of the polygon changes. If Antoine and Bobby got different images, one of them made a mistake in applying the rotation rules. A correct rotation of a pentagon about a point $P$ should follow the rigid - motion rules where distances and angle measures are preserved.

Step3: Answer question 1

Without seeing the actual rotations done by Antoine and Bobby, we assume that the rotation of a polygon about a point should be a rigid transformation. If the images are different, one of the rotations is incorrect because in a proper rotation (a rigid transformation), the shape and size of the polygon remain the same, and only its orientation changes.

Step4: Answer question 2

All four figures (after $90^{\circ},180^{\circ},270^{\circ}$ clock - wise rotations of rectangle $STUV$ about the origin) are congruent to the original rectangle. They all have the same side lengths and angle measures. The properties that they all have in common are:

• Angle measures: All angles in a rectangle are $90^{\circ}$, and after rotation, the angle measures remain $90^{\circ}$.
• Side lengths: The lengths of the sides of the rectangle do not change under rotation. Rotation is a rigid transformation that preserves distance (side - length) and angle measure.

Answer:

1. One of the rotations is likely incorrect. In a proper rotation (a rigid transformation), the shape and size of the pentagon should remain the same, and only its orientation should change. If the images are different, a mistake was made in applying the rotation rules.
2. They all have the same angle measures (all angles in a rectangle are $90^{\circ}$ and remain so after rotation) and the same side lengths (since rotation is a rigid transformation that preserves distance).