When you hit a pool ball that bounces off a bumper and into a pocket, that ball travels the shortest possible distance to the bumper and then to the pocket (assuming no "english" is placed on the ball and assuming it goes in.) Light bouncing off a mirror likewise travels the shortest possible distance. When you look at your feet in the mirror, light from your feet travels the shortest possible distance from your feet to the mirror to your eyes. Where is the point located on a mirror where a light ray reflects to minimize these distances?
Sketch
Step 1: Construct vertical segment AB.
Step 2: Construct segments CD and DE, where D is on segment AB.
Step 3: Measure segments CD and DE and calculate their sum. Make sure this sum is displayed to the thousandths precision.
Investigate
Move point D up and down the segment until segment CD + segment DE is minimized. Can you see any relationship between angle ADE and angle BDC? Measure these angles. Move C and E to different locations and minimize segment CD + segment DE again. Does the angle relationship still hold?
Conjecture: Write you conjectures below.
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Investigate Further
Some additional sketching will give you some ideas for why your conjecture is true. Mark segment AB as mirror in the Transform menu. Reflect C and segment CD across segment AB. How are lengths CD and C'D related? Move point D so that segment CD + segment DE is minimized. What can you say about C'D and E? What abou angles CDB, C"DB, and ADE? On a separate piece of paper, write an explanation for why your conjecture is true.
Hide segments CD, C'D, DE, and point D. Can you now construct a point F on segment AB such that CF + FE is minimal?
Explore More
What's the shortest mirror you'd need on a wall in order to see your full reflection from your toes to the top of your head?
Jackiw, Nicholas. The Geometer's Sketchpad, Version 3.05. Berkeley, CA: Key Curriculum Press.: 1996.
Key Curriculum Press. Exploring Geometry with The Geometer's Sketchpad. Berkley, CA:1996. p. 91.