Question

check understanding

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 angle measures, and their sides are parallel.

Explanation:

1.

Step1: Recall rotation properties

A rotation is a transformation that turns a figure around a fixed - point (the center of rotation). For a regular pentagon, the angle of rotational symmetry is $\frac{360^{\circ}}{n}$, where $n = 5$ (number of sides), so the angle of rotational symmetry is $72^{\circ}$. When rotating a pentagon about a point, the result depends on the center of rotation and the angle of rotation. If Antoine and Bobby rotated the pentagon about point $P$, as long as they used the same angle of rotation, they should get the same image. If they got different images, it is likely that they used different angles of rotation.

Step1: Recall rotation rules for a point $(x,y)$

The rotation rules about the origin are:

• For a $90^{\circ}$ clock - wise rotation: $(x,y)\to(y, - x)$
• For a $180^{\circ}$ rotation: $(x,y)\to(-x,-y)$
• For a $270^{\circ}$ clock - wise rotation: $(x,y)\to(-y,x)$

Let the vertices of rectangle $STUV$ be $S(x_1,y_1)$, $T(x_2,y_2)$, $U(x_3,y_3)$, $V(x_4,y_4)$.
For a $90^{\circ}$ clock - wise rotation about the origin:
If $S(x_1,y_1)$, then $S'(y_1,-x_1)$; if $T(x_2,y_2)$, then $T'(y_2,-x_2)$; if $U(x_3,y_3)$, then $U'(y_3,-x_3)$; if $V(x_4,y_4)$, then $V'(y_4,-x_4)$
For a $180^{\circ}$ rotation about the origin:
If $S(x_1,y_1)$, then $S'(-x_1,-y_1)$; if $T(x_2,y_2)$, then $T'(-x_2,-y_2)$; if $U(x_3,y_3)$, then $U'(-x_3,-y_3)$; if $V(x_4,y_4)$, then $V'(-x_4,-y_4)$
For a $270^{\circ}$ clock - wise rotation about the origin:
If $S(x_1,y_1)$, then $S'(-y_1,x_1)$; if $T(x_2,y_2)$, then $T'(-y_2,x_2)$; if $U(x_3,y_3)$, then $U'(-y_3,x_3)$; if $V(x_4,y_4)$, then $V'(-y_4,x_4)$

Step2: Analyze common properties

All four figures (the original rectangle and its three rotated images) are rectangles. Rectangles have opposite sides parallel, all four angles are right - angles ($90^{\circ}$), and opposite sides are equal in length.

Answer:

They are likely not correct because they probably used different angles of rotation about point $P$.

2.