Back to Exercise: Describe transformation effects

Exercises: Symmetries of Geometric Figures

Work through each section in order. Use precise transformation language in your explanations — state angles of rotation and lines of reflection exactly.

Grade 9·21 problems·~30 min·Common Core Math - HS Geometry·standard·hsg-co-a-3

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Warm-Up: Transformations Review

These problems review transformation concepts you have already studied.

A rigid motion is applied to a figure. Which property is preserved?

Position of the figure

Both distance between points and angle measures

Only angle measures, not distances

Only distances, not angle measures

Which statement about the identity transformation is correct?

The $360^\circ$ rotation is a different transformation from the identity.

The identity maps every point to itself and is always a symmetry of every figure.

The identity is not considered a symmetry because it does not move any points.

The identity is a translation by zero units.

In your own words, explain what it means for a rigid motion to "carry a figure onto itself." Use the phrase "maps the figure to itself" in your answer.

Fluency Practice

Identify the rotational and reflective symmetries of each figure.

$A regular hexagon with vertices labeled 1 through 6 and center marked.$

A regular hexagon is centered at the origin. Which list shows all distinct rotational symmetry angles (in degrees, greater than $0^\circ$ and less than $360^\circ$ )?

$60^\circ$ only

$60^\circ$ , $120^\circ$ , $180^\circ$ , $240^\circ$ , $300^\circ$ , $360^\circ$

$60^\circ$ , $120^\circ$ , $180^\circ$ , $240^\circ$ , $300^\circ$

$90^\circ$ , $180^\circ$ , $270^\circ$

A regular polygon has rotational symmetry at every multiple of $45^\circ$ (starting from $45^\circ$ , up to but not including $360^\circ$ ). How many sides does this regular polygon have?

A regular pentagon is centered at a fixed point. What is the smallest positive angle of rotation (in degrees) that maps the pentagon onto itself?

$Rectangle with its diagonal, horizontal midline, and vertical midline drawn as dashed lines.$

A non-square rectangle has vertices at $(0,0)$ , $(6,0)$ , $(6,2)$ , and $(0,2)$ . Which of the following is a line of reflective symmetry for this rectangle?

The diagonal from $(0,0)$ to $(6,2)$

The line $y = 1$ (horizontal midline)

The line $y = x$

Any line through the center $(3,1)$

$A non-rectangular parallelogram with labeled vertices and no lines of symmetry shown.$

A parallelogram has vertices at $(0,0)$ , $(4,0)$ , $(5,2)$ , and $(1,2)$ . This is a non-rectangular parallelogram (adjacent sides have different lengths and angles are not right angles). How many lines of reflective symmetry does it have?

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