What is the smallest a flat mirror can be while still being able to see your full reflection?

I need to show my work/explain. Can anyone help me?

draw a ray diagram from the eyes to the feet. Draw a second ray diagram from the eye to the top of the head. Now, use geometry and congruent triangles to solve your question.

Draw the mirror opposite you and parallel to you

Put your eye at the top of your head.
Draw a line from your eye straight to the mirror perpendicular to it. The line bounces right back to your eye, so you have the top of you.
Now draw a line from your eye, bounce off the mirror, and hit your foot.
You will notice the bounce occurs halfway down the mirror. By pretending the mirror has a hole at exactly that point halfway down, you can continue that line from your eye through the mirror through to the floor behind the mirror. Then you can work the geometry out by similar triangles if you wish.
So the mirror only needs to be half your height

Sure! To determine the smallest size of a flat mirror that allows you to see your full reflection, we need to consider the concept of angular resolution.

First, let's define angular resolution. It refers to the ability of our eyes to distinguish between two separate points or objects. The smaller the angular resolution, the smaller the objects we can differentiate.

To calculate the angular resolution, we can use the formula:
Angular Resolution = 1.22 * (wavelength of light) / (diameter of the mirror)

The wavelength of visible light is around 550 nanometers (nm) or 5.5 x 10^-7 meters.

To see your full reflection, you need to be able to differentiate between the top of your head and the bottom of your chin, which is roughly about 2 meters tall.

Let's assume that your eyes are 10 centimeters (0.1 meters) apart from each other when you stand in front of the mirror.

Now, we can calculate the diameter of the mirror using the formula:
Diameter of the mirror = 1.22 * (wavelength of light) / (Angular Resolution)

Consider that the angle formed between your two eyes and the top of your head is half of your total field of view. Therefore, the angular resolution should be equal to half of this angle.

Angular Resolution = (Height of the object) / (Distance to the object)

So, we can substitute the values into the formula:
Angular Resolution = 2 meters / 0.1 meters = 20 radians

Substituting the known values into the diameter formula:
Diameter = 1.22 * (5.5 x 10^-7 meters) / (20 radians)

By solving this equation, you can find the smallest size of the mirror that allows you to see your full reflection.