Faces, Vases and a Mirror

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In my geometry course, we have a mixed grade class. This means we have 9th to 12th graders in one geometry class. Since we are all in different grades, my teacher Mr. W, picked a problem that could be interpreted across many different skill levels. The question was, where can we find reflected angles in real life? How can we represent them? We are doing an action project because we believe that my class does not just need identify and quantify reflections using formal mathematical concepts, but relate it to real life concept.

My first thing to answer in this post is, what is a reflection? A reflection is a type of rigid transformation in which the pre-image is flipped across a line of reflection to create the image. Each point of the image is the same distance from the line as the pre-image is, just on the opposite side of the line. I picked the face/vase illusion. Automatically I noticed this picture is a reflection of itself vertically, so I decided to challenge myself and find any other data. I used Pythagorean theorem, slope and angles to help me calculate and represent reflected angles in real life.

C.H. (2018) Face and Vase Illusion. Web: Geogebra


The angle of ∠AFC is a complementary angle, at about 102˙. The angle of ∠BED is a complementary angle, at about 100˙. The slope of the line ⌳AF = 1 / 1. The slope of ⌳CF ≃ - 1.5. The slope of the line ⌳BE ≃ - 1. The slope of the line ⌳ED ≃ 1.5. The perimeter of the triangle △DEB = 13.79. The length of line DB = 6. This helps because I can use the perimeter and the length of the line DB to help me find the length of the line EB and ED. The length of line FA is ≃ 3.3, the length of the line CF is ≃ 4.3. With these two lengths I can use the formula C = a^2 + b^2 to help me find the hypotenuse. 


In conclusion, I found that using different coordinates and lines helped me to see the different ways I can make a reflection. By graphing different coordinates, angles, slope, and Pythagorean theorem. I was able to understand my reflections in a deeper sense. This was a fun process and I hope to continue learning new things throughout this course!

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