Question

1. regular pentagon qrstu is shown, where p is the center. which transformation carries qrstu onto itself?

a) a reflection over the x - axis
b) reflection over the line y = 1
c) rotation of 108° about point p
d) rotation of 144° about point p

1. the shaded piece has been transformed into a frieze pattern. which transformations best describe how the pattern was created?

a) rotations only
b) reflection only
c) translation and reflection
d) rotation and reflection

1. john eason has corners for a picture frame. which transformation could he use to show corners 1 and 2 are congruent?

a) reflection across line m
b) reflection across line n
c) translation to the right
d) rotation 180 degrees clockwise

Explanation:

Step1: Recall rotation property of regular pentagon

The central angle of a regular pentagon is $\frac{360^{\circ}}{n}$, where $n = 5$. So the central - angle is $\frac{360^{\circ}}{5}=72^{\circ}$. A rotation of $k\times72^{\circ}$ about the center of a regular pentagon will map the pentagon onto itself. When $k = 2$, the rotation angle is $144^{\circ}$. A reflection over the x - axis or the line $y = 1$ will not map the given pentagon onto itself as the orientation of the pentagon will change. A rotation of $108^{\circ}$ is not a multiple of $72^{\circ}$.

Step2: Analyze frieze - pattern transformation

A frieze pattern is created by a combination of translations (sliding the shape along a straight line) and reflections (flipping the shape over a line). Rotations alone or reflections alone will not create a frieze pattern like the one shown.

Step3: Determine congruence transformation for picture - frame corners

A reflection across line $n$ will map corner 1 onto corner 2, showing that they are congruent. A reflection across line $m$ will not map corner 1 to corner 2. A translation to the right will not map corner 1 to corner 2 as the orientation needs to be flipped. A $180^{\circ}$ clock - wise rotation will not map corner 1 to corner 2.

Answer:

1. D. rotation of 144° about point P
2. C. Translation and Reflection
3. B. Reflection across line n