Question

Octagon PQRSTVWZ is a regular octagon with its center at point C.

Which transformations will map octagon PQRSTVW Z onto itself?
Select each correct transformation

See attached for figure and choices.

Answer 1

Answer:

Option F is correct, i.e. "rotating 90° counterclockwise around point C."

Step-by-step explanation:

Given is the regular octagon PQRSTVWZ with its center at point C.

To map octagon PQRSTVWZ onto itself, we must provide such a transformation that should not change the size or coordinates of the vertices.

We can not use Translations, Reflections, and Dilations because they will change size and coordinates.

We can use only Rotations and that should be about the center C of the regular shape.

About center point C, and rotations of 90° does not change the orientation of the regular octagon, Both images would overlap each other.

Hence, option F is correct, i.e. "rotating 90° counterclockwise around point C."

Answer 2

The octagon can be reflected about a line of symmetry or rotated about an

angle of rotational symmetry to map it onto itself.

The transformations that will map octagon PQRSTVWZ onto itself are;

• Reflecting over  $\overline{QV}$

• Reflecting over $\overline{RW}$

• Rotating 135° clockwise around point C

• Rotating 90° counterclockwise around point C

Reason:

The angle formed at the center by a side of the hexagon is $\dfrac{360^{\circ}}{8} = 45^{\circ}$and a reflection about each diagonal maps the octagon onto itself, which corresponds with the properties of an octagon including;

• An octagon has 8 lines of symmetry which are the diagonals

• An octagon has a rotational symmetry of order 8, including, 45°, 90°, 135°, 180°, 225°, 270°, 315°, and 360°, about the center.

• A reflection about a diagonal and a rotation about the center C, by an angle equal to an angle of rotational symmetry, will map the octagon onto itself,  therefore;

The transformations that will map octagon PQRSTVWZ onto itself are;

Reflection across the diagonal $\overline{QV}$

Reflection across the diagonal $\overline{RW}$

Rotation 135° clockwise around point C

Rotation 90° counterclockwise around point C

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