Question

question 6/25
6 octagon pqrstvwz is a regular octagon with its center at point c.
which transformations will map pqrstvwz onto itself?
select each correct transformation.
a reflecting over \\(\overline{rw}\\)
b reflecting over \\(\overline{qv}\\)
c reflecting over \\(\overline{tz}\\)
d rotating \\(135^\circ\\) clockwise around point c
e rotating \\(90^\circ\\) counterclockwise around point c.
f rotating \\(45^\circ\\) clockwise around point z

Explanation:

Step1: Identify symmetry axes

A regular octagon has 8 lines of symmetry: 4 pass through opposite vertices, 4 pass through midpoints of opposite sides.

• $\overline{RW}$: Connects vertex R to vertex W (opposite vertices) → symmetry axis.
• $\overline{QV}$: Connects vertex Q to vertex V (opposite vertices) → symmetry axis.
• $\overline{TZ}$: Connects vertex T to vertex Z (not opposite vertices) → not a symmetry axis.

Step2: Analyze rotational symmetry

The central angle between adjacent vertices is $\frac{360^\circ}{8}=45^\circ$. Rotations by multiples of $45^\circ$ around center C map the octagon to itself.

• $135^\circ = 3\times45^\circ$ (multiple of $45^\circ$) → valid rotation.
• $90^\circ = 2\times45^\circ$ (multiple of $45^\circ$) → valid rotation.
• Rotation around Z (not center) → does not map the octagon to itself.

Answer:

A. reflecting over $\overline{RW}$
B. reflecting over $\overline{QV}$
D. rotating $135^\circ$ clockwise around point $C$
E. rotating $90^\circ$ counterclockwise around point $C$