IntroductionA line of symmetry, , is a line separating a figure into two halves that are mirror images. Line symmetry exists for a figure if for every point P on one side of the line, there is a corresponding point Q where is the perpendicular bisector of .

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5.1.3: Applying Lines of Symmetry

Introduction, continuedFrom the diagram, we see that is perpendicular to . The tick marks on the segment from P to R and from R to Q show us that the lengths are equal; therefore, R is the point that is halfway between .

Depending on the characteristics of a figure, a figure may contain many lines of symmetry or none at all. In this lesson, we will discuss the rotations and reflections that can be applied to squares, rectangles, parallelograms, trapezoids, and other regular polygons that carry the figure onto itself. Regular polygons are two-dimensional figures with all sides and all angles congruent.

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5.1.3: Applying Lines of Symmetry

Introduction, continuedSquares

Because squares have four equal sides and four equal angles, squares have four lines of symmetry. If we rotate a square about its center 90˚, we find that though the points have moved, the square is still covering the same space.

Similarly, we can rotate a square 180˚, 270˚, or any other multiple of 90˚ with the same result.

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5.1.3: Applying Lines of Symmetry

Introduction, continuedWe can also reflect the square through any of the four lines of symmetry and the image will project onto its preimage.

Rectangles

A rectangle has two lines of symmetry: one vertical and one horizontal. Unlike a square, a rectangle does not have diagonal lines of symmetry. If a rectangle is rotated 90˚, will the image be projected onto its preimage? What if it is rotated 180˚?

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5.1.3: Applying Lines of Symmetry

Introduction, continued

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5.1.3: Applying Lines of Symmetry

Introduction, continuedIf a rectangle is reflected through its horizontal or vertical lines of symmetry, the image is projected onto its preimage.

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5.1.3: Applying Lines of Symmetry

Horizontal reflection

A, B'

B, A'

D, C' C, D'

Vertical reflection

A, D' B, C'

D, A' C, B'

Introduction, continuedTrapezoids

A trapezoid has one line of symmetry bisecting, or cutting, the parallel sides in half if and only if the non-parallel sides are of equal length (called an isosceles trapezoid). We can reflect the isosceles trapezoid shown below through the line of symmetry; doing so projects the image onto its preimage. However, notice that in the last trapezoid shown on the next slide, is longer than , so there is no symmetry. The only rotation that will carry a trapezoid that is not isosceles onto itself is 360˚.

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5.1.3: Applying Lines of Symmetry

Introduction, continued

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5.1.3: Applying Lines of Symmetry

Introduction, continuedParallelograms

There are no lines of symmetry in a parallelogram if a 90˚ angle is not present in the figure. Therefore, there is no reflection that will carry a parallelogram onto itself. However, what if it is rotated 180˚?

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5.1.3: Applying Lines of Symmetry

Introduction, continued

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5.1.3: Applying Lines of Symmetry

Key Concepts• Figures can be reflected through lines of symmetry

onto themselves.

• Lines of symmetry determine the amount of rotation required to carry them onto themselves.

• Not all figures are symmetrical.

• Regular polygons have sides of equal length and angles of equal measure. There are n number of lines of symmetry for a number of sides, n, in a regular polygon.

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5.1.3: Applying Lines of Symmetry

Common Errors/Misconceptions• showing a line of symmetry in a parallelogram or

rhombus where there isn’t one

• missing a line of symmetry

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5.1.3: Applying Lines of Symmetry

Guided Practice

Example 1Given a regular pentagon ABCDE, draw the lines of symmetry.

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5.1.3: Applying Lines of Symmetry

Guided Practice: Example 1, continued

1. First, draw the pentagon and label the vertices.

Note the line of symmetry from A to .

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5.1.3: Applying Lines of Symmetry

Guided Practice: Example 1, continued

2. Now move to the next vertex, B, and extend a line to the midpoint of .

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5.1.3: Applying Lines of Symmetry

Guided Practice: Example 1, continued

3. Continue around to each vertex, extending a line from the vertex to the midpoint of the opposing line segment.

Note that a regular pentagon has five sides, five vertices, and five lines of reflection.

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5.1.3: Applying Lines of Symmetry

✔

Guided Practice: Example 1, continued

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5.1.3: Applying Lines of Symmetry

Guided Practice

Example 3Given the quadrilateral ABCE, the square ABCD, and the information that F is the same distance from A and C, show that ABCE is symmetrical along .

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5.1.3: Applying Lines of Symmetry

Guided Practice: Example 3, continued

1. Recall the definition of line symmetry.Line symmetry exists for a figure if for every point on one side of the line of symmetry, there is a corresponding point the same distance from the line.

We are given that ABCD is square, so we know .

We also know that is symmetrical along .

We know . 19

5.1.3: Applying Lines of Symmetry

Guided Practice: Example 3, continued

2. Since and , is a line of symmetry for where .

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5.1.3: Applying Lines of Symmetry

Guided Practice: Example 3, continued

3. has the same area as because they share a base and have equal height. , so .

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5.1.3: Applying Lines of Symmetry

Guided Practice: Example 3, continued

4. We now know is a line of symmetry

for

and is a line of symmetry for

, so and

quadrilateral ABCE is symmetrical along

.

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5.1.3: Applying Lines of Symmetry

✔

Guided Practice: Example 3, continued

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5.1.3: Applying Lines of Symmetry