Congruence: Symmetries of Polygons: Rotations and Reflections (CCSS.G-CO.3)
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Common Core High School Geometry › Congruence: Symmetries of Polygons: Rotations and Reflections (CCSS.G-CO.3)
Parallelogram $ABCD$ with vertices $A(0,0)$, $B(6,0)$, $C(8,4)$, $D(2,4)$.
Which rigid motion carries this figure onto itself?
Reflection across the line $y=2$
Reflection across the diagonal $\overline{AC}$
Rotation by $180^\circ$ about the midpoint of $\overline{BD}$
Rotation by $90^\circ$ about the origin
Explanation
A generic parallelogram (not a rectangle or rhombus) has exactly one symmetry: a rotation by $180^\circ$ about the intersection of its diagonals. Reflecting across $y=2$ (choice A) does not map $A(0,0)$ to $D(2,4)$, so it is not a symmetry; reflecting across a diagonal (choice B) is also not a symmetry of a general parallelogram.
A regular hexagon with side length 4 units.
How many lines of symmetry does this polygon have?
6
3
12
2
Explanation
A regular $n$-gon has $n$ lines of symmetry, so a regular hexagon has 6. Choice C (12) doubles the count by confusing lines of symmetry with rotational symmetries.
An isosceles trapezoid with vertices $(-5,0)$, $(5,0)$, $(3,4)$, $(-3,4)$ (bases are horizontal).
Which rigid motion carries this figure onto itself?
Reflection across the line $y=2$
Rotation by $180^\circ$ about $(0,2)$
Reflection across the diagonal $\overline{(-5,0)(3,4)}$
Reflection across the line $x=0$
Explanation
An isosceles trapezoid has exactly one line of symmetry: the perpendicular bisector of the bases, here the vertical line $x=0$. Choice A (reflecting across $y=2$) would swap the long base with the short base and does not preserve the shape.
Rectangle with vertices $(-4,-2.5)$, $(4,-2.5)$, $(4,2.5)$, $(-4,2.5)$ (not a square).
How many lines of symmetry does this polygon have?
0
2
4
1
Explanation
A non-square rectangle has 2 lines of symmetry: the lines through the center parallel to the sides (here $x=0$ and $y=0$). Choice C (4) incorrectly includes the diagonals, which are lines of symmetry only for a square.
A regular pentagon with side length 6 units.
Which rigid motion carries this figure onto itself?
Rotation by $90^\circ$ about its center
Reflection across a diagonal connecting two nonadjacent vertices
Rotation by $72^\circ$ about its center
Rotation by $120^\circ$ about its center
Explanation
A regular pentagon has rotational symmetry through multiples of $72^\circ$, so a $72^\circ$ turn about the center maps it onto itself. Choice B is incorrect because the axes of reflection symmetry go through a vertex and the midpoint of the opposite side, not along diagonals.
A regular hexagon has its center at (0,0) and all vertices on a circle of radius 6. Which rigid motion carries this figure onto itself?
Rotate 60° about the center
Rotate 45° about the center
Reflect across a diagonal
Rotate 90° about the center
Explanation
A regular hexagon has rotational symmetry of $360^\circ/6=60^\circ$, so rotating $60^\circ$ about the center maps it onto itself. Choice D (90°) is not a multiple of $60^\circ$, so it does not return the hexagon to its original position.
Parallelogram with vertices A(0,0), B(6,0), C(8,4), and D(2,4). Which rigid motion carries this figure onto itself?
Reflect across diagonal AC
Rotate 180° about the intersection of the diagonals
Reflect across the horizontal line y = 2
Rotate 90° about the intersection of the diagonals
Explanation
Any parallelogram has 180° rotational symmetry about the intersection point of its diagonals, mapping each vertex to the opposite vertex. Choice A is incorrect: reflecting across a diagonal of a general parallelogram does not produce the same figure unless it is a special case (like a rhombus or rectangle with extra symmetry).
A regular pentagon has side length 4 units. How many lines of symmetry does this polygon have?
5
10
2
4
Explanation
A regular $n$-gon has $n$ lines of symmetry, each passing through a vertex and the midpoint of the opposite side, so a regular pentagon has 5. Choice B (10) double-counts diagonals, which are not all symmetry axes.