If you have ever seen an ophthalmologist for an eye exam, you have probably had your pupils dilated. It is a strange sensation that makes you very sensitive to light, but it also allows your eye doctor to see inside your eye. When you get your pupils dilated, it changes the size of your pupils from a smaller to a bigger hole.
Dilation plays a role in geometry problems when the size of a figure changes, but the shape does not. To remember what dilation means in geometry, think of soccer balls that are re-sized, or "dilated," as players move into older age groups. Watch this video to learn more about the kind of dilation that appears in geometry problems.
As you watch this video, use the study guide to follow along if you'd like. Click the button below to download the study guide.
What does a visit to the eye doctor have to do with Geometry? If you remember, in Geometry, we say that similar
shapes are objects that are the same shape, but a different size. If you go to the eye doctor and have only one pupil
dilated, or if you were a pirate voluntarily wearing an eye patch, your pupils will be the same shape, a circle, but
different sizes. But if both eyes were to dilate to the same shape and size, your pupils would NOT be similar, but actually
congruent.
What does a visit to the eye doctor have to do with Geometry? If you remember, in Geometry, we say that similar
shapes are objects that are the same shape, but a different size. If you go to the eye doctor and have only one pupil
dilated, or if you were a pirate voluntarily wearing an eye patch, your pupils will be the same shape, a circle, but
different sizes. But if both eyes were to dilate to the same shape and size, your pupils would NOT be similar, but actually
congruent.
What about these next objects? This time I selected a few interesting points on each soccer ball and showed the
distances between each. Based on the information provided would you say these are similar? How can you tell? If so,
what is the scale factor to complete the dilation from the ball on the left to the right? Pause the video and answer the
questions on the screen… Yes these are similar; they are the same shape, but different sizes. The scale factor is one
half.
What about these images of the Lincoln Memorial. Are they similar? How can you tell? And what's the scale factor?
Since no measurements are given, you might want to pick a few points on the diagram and determine their distances
based on the grid system provided. Pause the video and prepare your answers. Check your solutions in just a moment
by resuming the video… I chose the following points on both images and calculated distances between them three
different ways by using the coordinates and the distance formula. Since their distances are all the same, these objects
are NOT similar, but rather they are congruent – they are the same shape AND size. Therefore, the scale factor is not
applicable here.
Let's talk a little bit about how to draw a geometric dilation. If you are provided with an image that needs to be dilated,
you can do so in three steps. First, pick a representative number of points on a coordinate grid. In this case, I'll just use
the three vertices of the triangle. If we were working with the soccer ball or Lincoln monument examples, we might only
select a few points as illustrated earlier. Next, multiply the coordinates of these points by your scale factor. And finally
plot the new points. I'd like for you to sketch this triangle dilated by a scale factor of three. Please pause the video and
complete the drawing. Check your work in just a moment by resuming the video… And there you have it. Each
coordinate, the x- and y- component, was multiplied by three and the new coordinates were plotted.
Feel free to try this technique on your own by creating your own coordinate grid around an image if it's not already
provided. Good luck!
Question
Why does a dilation produce similar figures but not congruent figures?
