My friend sent me this problem, and my efforts to solve it have thus far been frustrated. I need some insight!
Joe is 6 feet tall, and standing in front of a mirror that is at eye level, and is 3 feet tall. He is holding a model person 3 inches tall 1 foot in front of his eyes. If he perceives the model and his reflection appear to be the exactly same size, how far is he standing from the mirror?
I get a vague feeling that the intercept theorem can be used here, but am unsure how to model his model and reflection appearing equal mathematically.
A crude diagram:
NOTE
"eye level" is taken to mean the middle of the mirror is parallel to Joe's line of sight i.e. the diagram is NOT to scale.
Construct a simple ray diagram, the mirror bisects all rays from Joe's eye to Joe's image. Vertical displacement is not important.
By virtue of similar triangles:
Ratio of image heights = $ \dfrac{6^{'}}{3^{"}}=24 $
Distance to image = $ 1^{'} \cdot 24 =24 ^{'}$
Distance to image is one half, = $ 12^{'}.$