Reflections, Glide Reflections, and Dilations

nataliehobson

2 Reflections, Glide Reflections, and Dilations

Learning Objectives

Understand and use reflections
Understand and use dilations

Reflection

Another type of isometry is a reflection. The most common example of a reflection often encountered in our daily lives is our image in a mirror.

We can obtain reflections about a line in various ways. Folding the paper along thereflecting line(in blue) and drawing the image gives the mirror image, or image, of the figure.

Tracing paper can be used. Trace the original figure, the reflecting line, and a point on the reflecting line, which we use as a reference point, first picture. Flip the tracing paper over to perform the reflection, and align the reflecting line and the reference point, second picture.

Reflections in a Coordinate System

Example- Reflection over the line y = x

Under the reflection over the line y = x, a coordinate point (s, t) goes to the point (t, s).

Practice Exercises

Solution: The image is the same as the preimage.

Solution: Using a tape measure or ruler, show that the distance from A perpendicular to the line is the same as the distance from A’ perpendicular to the line. Do the same with B and B’, C and C’.

3. Find the equation of the images of the following lines when the reflection line is the x-axis.

Solution: Red line

Solution: Blue line

Glide Reflection

A glide reflection, another basic isometry, is a transformation consisting of a translation (green arrow), followed by a reflection in a line (blue arrow) parallel to the slide arrow (orange arrow).

Thus the boat is moved forward and flipped over.

Dilation

A dilation is a transformation that resizes the object, smaller to a reduction or larger to an enlargement. The scale factor represents the amount of the resizing.

Below are two examples of images of a dilation. Sometimes a point outside the object (e.g., O in the dilation on the left below) is used to assist in the alignment. The diagram on the right uses the xy-coordinate plane to correctly line up the enlargement (red triangle to blue triangle) or reduction (blue triangle to red triangle). Can you find the scale factor for both of these dilations?

O