Optics - Finding the height of a mirror

[Xyius]

Homework Statement

Determine the minimum height of a wall mirror that will permit a 6ft person to view his or her entire height. Sketch rays from the top and bottom of the person, and determine the proper placement of the mirror such that the full image is seen, regardless of the persons distance from the mirror.

Homework Equations

The Law of reflection and refraction.

The Attempt at a Solution

I do not know where to start with this one. The question itself confuses me, how can the full image always be seen regardless of the distance from the mirror? What if the mirror is right up against the person? How could he possibly see his whole image? The solution in the back of the book says, "3ft with top edge of mirror at a height halfway between the persons eye level and the top of the person's head." How in the world would I go about getting this?

 

[vk6kro]

They are talking about the height of the mirror from top to bottom of the mirror, not height from the ground.

Draw a line from your feet to the mirror and then it reflects towards your eyes. What do you know about the angle of incidence and the angle of reflection?

Similarly, draw a line from the top of your head, reflecting off the mirror and hitting your eyes.
What do you know about the angle of incidence and the angle of reflection?

 

[Xyius]

I know that in each case, the angle of incidence will equal the angle of reflection. I am confused about getting the height of the mirror. I am looking to find some sort of geometric approach where "h" (the height of the mirror) is the height of a triangle. Nothing is immediately obvious to me though :\

 

[Xyius]

Oh! I think I got it! (But please correct me if I am wrong)

Drawing a line from the very bottom of my foot to the mirror and then to my eye, it is evident that any "mirror" below that point is not needed, therefore the light ray from my foot hits the very bottom of the mirror.

Since the angle of incidence equals the angle of reflection, it forms two triangles back to back with me acting as the base of the two triangles. (Hard to explain without a picture, basically 6 feet is the sum of both the triangle bases.) Since the two angles are the same by the mirror, and each triangle has a 90 degree angle in it, the two triangles are the same and divide 6 feet evenly, meaning the mirror would be 3 feet!

 

[vk6kro]

Yes, that's right. Good.

It would be even easier if you had eyes at the top of your head, but it is pretty easy anyway.

Just drawing a large, accurate drawing usually gets you half way towards an answer in optics.

 

[Xyius]

Cool! Thanks!