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·Common Core Math - HS Geometry·standard·hsg-co-a-3

Work through problems with immediate feedback

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 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?

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 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?

Mixed Practice

These problems present symmetry concepts in different formats.

Complete the symmetry profile. A regular $n$ -gon has ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ rotational symmetries (including the identity) and ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ lines of reflective symmetry.

number of rotational symmetries:

number of lines of reflective symmetry:

A regular polygon has exactly 8 lines of reflective symmetry. At which of the following angles does it NOT have rotational symmetry?

$45^\circ$

$90^\circ$

$135^\circ$

$50^\circ$

The figure below shows a square with four candidate fold lines (dashed). Which of the following is NOT a line of reflective symmetry of the square?

m1 (horizontal midline)

m2 (vertical midline)

The line connecting the midpoints of opposite sides at a $45^\circ$ angle to m1

m3 (diagonal from A to C)

An isosceles trapezoid has ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ line(s) of reflective symmetry and ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ non-trivial rotational symmetry (rotational symmetries other than the identity).

number of lines of symmetry:

number of non-trivial rotational symmetries:

A student claims: "If a figure has rotational symmetry, it must also have at least one line of reflective symmetry." Is this claim true or false? Give a specific figure as evidence and explain your reasoning.

Word Problems

Read each problem carefully and apply symmetry concepts to answer.

A designer is creating a tile pattern using regular octagons. She wants to know how many distinct positions she can rotate one octagon tile (about its center) and have it look identical to its starting position. She counts the identity position as one of them.

How many distinct rotation positions map the regular octagon onto itself?

A decorative flag is shaped like an isosceles trapezoid. The two parallel sides (bases) have lengths 10 cm and 6 cm. The two non-parallel sides (legs) each have length 5 cm.

How many lines of reflective symmetry does this flag shape have?

Which transformation describes the line of symmetry for this trapezoid?

Reflection across the perpendicular bisector of the two parallel sides

Rotation by $180^\circ$ about the center

Reflection across either diagonal

Reflection across a line connecting the midpoints of the two legs

A student compares the symmetries of a square and a non-square rectangle.

Explain why a square has more symmetries than a non-square rectangle. Reference both rotational and reflective symmetry in your answer.

Find the Mistake

Each problem shows a student's incorrect claim. Identify the error and explain why it is wrong.

Taylor solved this problem: "Find all lines of symmetry for a non-square rectangle with vertices at $(0,0)$ , $(4,0)$ , $(4,2)$ , $(0,2)$ ."

Taylor's answer: "The rectangle has 4 lines of symmetry: the horizontal midline $y = 1$ , the vertical midline $x = 2$ , and both diagonals from $(0,0)$ to $(4,2)$ and from $(4,0)$ to $(0,2)$ ."

What error did Taylor make, and why is it incorrect?

Taylor included the diagonals as lines of symmetry, but a diagonal of a non-square rectangle does NOT map the rectangle onto itself.

Taylor forgot to include the diagonals — a non-square rectangle has 6 lines of symmetry.

Taylor is correct — a non-square rectangle does have 4 lines of symmetry.

Taylor's error is counting the midlines — neither midline is a line of symmetry for a rectangle.

Morgan solved this problem: "How many lines of reflective symmetry does a non-rectangular parallelogram have?"

Morgan's answer: "A parallelogram has $180^\circ$ rotational symmetry, so it must have at least one line of reflective symmetry. I'll say it has 2 lines of symmetry — the two diagonals."

Identify the two errors in Morgan's reasoning.

Morgan correctly identified 2 lines of symmetry but used the wrong lines — they should be the midlines, not the diagonals.

Morgan made two errors: (1) concluding that rotational symmetry implies reflective symmetry (it does not), and (2) claiming the diagonals are lines of symmetry (they are not for a non-rectangular parallelogram).

Morgan's only error is claiming the diagonals are lines of symmetry — the conclusion that there is at least one line of symmetry is correct.

Morgan is correct — a non-rectangular parallelogram has 2 lines of symmetry (the diagonals).

Challenge Problems

These stretch problems require multi-step reasoning or explanation.

A regular decagon (10 sides) is centered at the origin. How many total symmetries does it have, counting both rotational symmetries (including the identity) and reflective symmetries?

A rhombus is a parallelogram with all four sides equal (but not necessarily right angles). Compare the symmetries of a rhombus (non-square) to those of a non-rectangular parallelogram. Explain what property of the rhombus gives it more symmetries.

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