In this video, I would like to show you ways to complete common transformations on a coordinate grid, and then talk a bit more about congruence versus similarity.

In this module, we discussed four different transformations: translations, reflections, rotations, and dilations. These transformations manipulate, or change, a shape from what we call its "pre-image" position to its "image" position. A translation slides the pre-image to the image, and produces two congruent shapes. To complete this on a coordinate grid, simply add or subtract values to existing coordinates. For example, if I want to slide this triangle six units to the right, and four units down, I would add six to each x-component of the original coordinates, and subtract four from each y-component. Pause the video and complete the translation. Resume the video in just a moment to check your work… This is what your image should look like based on the pre-image given and the translation described.

A reflection essentially flips a pre-image over a line to a new position and produces two congruent shapes. In order to reflect a shape over the x- or y-axis, all you need to do is change the signs of one of the components of the coordinates. For example, to reflect over the x-axis, change the sign of each y-component in the given coordinates and plot the new points. Pause the video now and complete this step. Check your work by resuming the video in just a moment… This is what your image should look like based on the pre-image given and the reflection described. Now reflect the original pre-image over the y-axis by applying the appropriate rule… Here is the pre-image reflected over the y-axis by changing the sign of the x-component of each coordinate.

A rotation turns a pre-image around a point and aligns it to a new image, congruent to the original. In order to complete this, start by switching the x- and y- component of a coordinate, and change one or both of the signs. For example, to rotate this triangle ninety degrees clockwise, I would switch the x- and y- components and change the sign of the new ycomponent like this. The coordinate negative 3, positive seven becomes positive 7, positive three. Try the other two on your own and plot the new coordinates. Check your answers in just a moment by resuming the video… Here is what the image should look like following this transformation.

The final transformation is the dilation. It changes a pre-image to an image by multiplying each component of every coordinate by a scale factor. This produces two similar shapes, not congruent, because the side lengths become proportional in size, and do not remain congruent. Multiply these coordinates by a scale factor of one-half, plot the points and observe the final image. Check your work by resuming the video in just a moment… This is what your image should look like. It may have been a bit difficult to graph, but manageable, due to the points not falling perfectly on a gridline.

To complete this video, I'd like to remind you of what to look for when proving two shapes are similar, rather than congruent. If two shapes were congruent, all pairs of angles and side lengths would be the exact same. Triangles, for example, would have six pairs of congruent parts. However, in order for two objects to be called similar, only the angles must be congruent; the side lengths can be proportional to one another. We can verify angles are congruent by their markings, we don't actually need to measure them. In the example below, the angle pairs match with thirty degrees, forty-five degrees, and one hundred five degrees respectively. Notice though, the angles measure weren't indicated, only the angle markings were given; showing pairs of congruent angles. The side lengths of the triangle on the right are three times the size of the triangle on the left. It might be difficult to recognize this because of the decimals, so we can check for proportionality, regardless of the scale factor, by organizing three fractions that should be equal to one another. Create your fractions by organizing corresponding side lengths, and cross multiply to check for equivalence. Let's write 3.845 over 11.535, 5.437 over 16.312, and 7.428 over 22.283. Cross multiply a numerator with a denominator for each pair of fractions: the product should be equal. Due to the measurements being rounded in the diagram, we will ignore slight differences in results due to rounding. Here you can see that within each pair of fractions, the cross products match. This indicates that all three pairs of sides are proportional in size. Since the side lengths are proportional and the angle measures are congruent, we know for certain that the two triangles are similar in shape. Please try this technique on your own before moving on to another section of this lesson. Feel free to search the internet by typing "proportions practice." Good luck!