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Precalculus: Help and Review11 chapters | 88 lessons

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Lesson Transcript

Instructor:
*Beverly Maitland-Frett*

Beverly has taught mathematics at the high school level and has a doctorate in teaching and learning.

Reflectional symmetry can be found in geometric figures, math, nature and the man-made world. In this lesson, we'll explore the characteristics that define reflectional symmetry, as well as some everyday examples.

What if you took a picture of yourself, particularly a passport type photograph, and drew a line straight down the middle of your face, from your forehead down to your chin? What would you notice? Wouldn't it seem as if one side of your face was a reflection of the other? For example, there would be an eye on each side. Both halves of your lips would look nearly identical. Unless you'd suffered some type of injury, both halves of your nose would look the same.

Ideally, your hypothetical passport photo is just one example of **reflectional symmetry**, also known as **bilateral**, **line**, or **mirror symmetry**. The line you drew to divide your face is called the **line of symmetry**.

However, since humans have uncontrollable differences, our faces may not always count as examples of reflectional symmetry. For instance, some of us may have a beauty spot on one side; others may have a scar. If you look closely in a mirror, you may notice that one of your eyes is a little smaller than the other. Many aspects of human appearance may distort the notion of true reflectional symmetry; therefore, reflectional symmetry must satisfy certain conditions.

**Reflectional symmetry** occurs when a line is used to split an object or shape in halves so that each half reflects the other half. Sometimes objects or shapes have more than one line of symmetry. Take, for instance, the letter H. How many lines of symmetry does it have? If you answered two, you are correct. There are two ways to draw a line so that each half reflects the other half.

Many letters of the alphabet have reflectional symmetry. Some use a vertical line; some use a horizonatal line. Geometric shapes can also demonstrate reflectional symmetry, such as circles and squares, which have four lines of symmetry. Depending on the type of triangle, one may have zero, one or three lines of symmetry.

While some shapes have one, two, or many lines of symmetry, some have none. Take the letter *N* for example, while it demonstrates point symmetry, it does not have reflectional symmetry. It is also possible for some shapes to have an infinite number of reflected images.

Reflectional symmetry can also be found in nature, such as the image of hills reflected in the water or snowflakes. Some members of the insect kingdom have reflectional symmetry, such as beetles, butterflies or flies. Many flowers also have reflectional symmetry.

Architects also use reflectional symmetry to beautify their buildings. Famous buildings, such as the Taj Mahal and the Eiffel Tower, have line or reflectional symmetry.

Some graphs have a line of symmetry where the sides to the left and right of the axis are identical to each other. Lines of symmetry can also be found in some trigonometric functions. For example, functions that even or odd are symmetric with the y-axis and origin respectively.

**Reflectional symmetry** is also called **bilateral**, **line symmetry** or **mirror symmetry**. It occurs when a line is drawn to divide a shape in halves so that each half is a reflection of the other. Some shapes or objects, such as circles, squares and triangles, have one or more lines of symmetry. Other shapes do not have line symmetry, but exhibit point symmetry. Reflectional symmetry can also be found in nature, such as in insects or mirror images in water. In math, some graphs and trigonometric functions also demonstrate reflectional symmetry.

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Precalculus: Help and Review11 chapters | 88 lessons

- What is a Parabola? 4:36
- Parabolas in Standard, Intercept, and Vertex Form 6:15
- What is a Function? - Applying the Vertical Line Test 5:42
- Multiplying Binomials Using FOIL and the Area Method 7:26
- How to Factor Quadratic Equations: FOIL in Reverse 8:50
- Factoring Quadratic Equations: Polynomial Problems with a Non-1 Leading Coefficient 7:35
- How to Complete the Square 8:43
- Completing the Square Practice Problems 7:31
- How to Solve a Quadratic Equation by Factoring 7:53
- How to Use the Quadratic Formula to Solve a Quadratic Equation 9:20
- How to Solve Quadratics That Are Not in Standard Form 6:14
- Graphing Circles: Identifying the Formula, Center and Radius 8:32
- Factoring Quadratic Expressions: Examples & Concepts
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