12.1: Exploring Symmetry
Learning Objectives
- Understand line and rotational symmetry.
Review Queue
- Define symmetry in your own words.
- Plot the points \begin{align*}A(1, 3), B(3, 1), C(5, 3),\end{align*} and \begin{align*}D(3, 5)\end{align*}.
- Find the slope of each side of the quadrilateral in #2.
- Find the slope of the diagonals of the quadrilateral. What kind of shape is this?
Know What? Symmetry exists all over nature. One example is a starfish. Determine if the starfish has line symmetry or rotational symmetery.
Lines of Symmetry
Line of Symmetry: A line that passes through a figure such that it splits the figure into two congruent halves.
Example 1: Find all lines of symmetry for the shapes below.
a)
b)
c)
d)
Solution: For each figure, draw lines through the figure so that the lines perfect cut the figure in half. Figure a) has two lines of symmetry, b) has eight, c) has no lines of symmetry, and d) has one.
a)
b)
c)
d)
Figures a), b), and d) all have line symmetry.
Line Symmetry: When a figure has one or more lines of symmetry.
These figures have line symmetry:
These figures do not have line symmetry:
Example 2: Do the figures below have line symmetry?
a)
b)
Solution: Yes, both of these figures have line symmetry.
a)
b)
Rotational Symmetry
Rotational Symmetry: When a figure can be rotated (less that \begin{align*}180^\circ\end{align*}) and it looks like it did before the rotation.
Center of Rotation: The point a figure is rotated around such that the rotational symmetry holds.
For the \begin{align*}H\end{align*}, we can rotate it twice, the triangle can be rotated 3 times and still look the same and the hexagon can be rotated 6 times.
Example 3: Determine if each figure below has rotational symmetry. Find the angle and how many times it can be rotated.
a)
b)
c)
Solution:
a) The pentagon can be rotated 5 times. Because there are 5 lines of rotational symmetry, the angle would be \begin{align*}\frac{360^\circ}{5} 72^\circ\end{align*}.
b) The \begin{align*}N\end{align*} can be rotated twice. This means the angle of rotation is \begin{align*}180^\circ\end{align*}.
c) The checkerboard can be rotated 4 times. There are 4 lines of rotational symmetry, so the angle of rotation is \begin{align*}\frac{360^\circ}{4}=90^\circ\end{align*}.
Know What? Revisited The starfish has 5 lines of symmetry and 5 lines of rotational symmetry. The angle of rotation is \begin{align*}72^\circ\end{align*}. The center of rotation is the center of the starfish.
Review Questions
- Questions 1-15 use the definitions of figures and symmetry.
- Questions 16-18 ask you to draw figures based on symmetry.
- Questions 19-38 are similar to Examples 1 and 3.
- Questions 39-41 are similar to Example 2.
Fill in the blanks.
- If a figure has 3 lines of rotational symmetry, it can be rotated _______ times.
- If a figure can be rotated 6 times, it has _______ lines of rotational symmetry.
- If a figure can be rotated \begin{align*}n\end{align*} times, it has _______ lines of rotational symmetry.
- To find the angle of rotation, divide \begin{align*}360^\circ\end{align*} by the total number of _____________.
- Every square has an angle of rotation of _________.
True or False
- All right triangles have line symmetry.
- All isosceles triangles have line symmetry.
- Every rectangle has line symmetry.
- Every rectangle has exactly two lines of symmetry.
- Every parallelogram has line symmetry.
- Every square has exactly two lines of symmetry.
- Every regular polygon has three lines of symmetry.
- Every sector of a circle has a line of symmetry.
- Every parallelogram has rotational symmetry.
- Every figure that has line symmetry also has rotational symmetry.
Draw the following figures.
- A quadrilateral that has two pairs of congruent sides and exactly one line of symmetry.
- A figure with infinitely many lines of symmetry.
- A figure that has one line of symmetry and no rotational symmetry.
Find all lines of symmetry for the letters below.
- Do any of the letters above have rotational symmetry? If so, which one(s) and what are the angle of rotation?
Determine if the words below have line symmetry or rotational symmetry.
- OHIO
- MOW
- WOW
- KICK
- pod
Trace each figure and then draw in all lines of symmetry.
Find the angle of rotation and the number of times each figure can rotate.
Determine if the figures below have line symmetry or rotational symmetry. Identify all lines of symmetry and the angle of rotation.
Review Queue Answers
- Where one side of an object matches the other side; answers will vary.
- \begin{align*}m_{\overline{AD}} = 1, m_{\overline{AB}} = -1, m_{\overline{BC}} = 1, m_{\overline{CD}} = -1\end{align*}
- \begin{align*}m_{\overline{AC}} = 0, m_{\overline{BD}} = undefined\end{align*}. The figure is a square.
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