Algebra of focus

okay now that you're comfortable with this diagram I think you are we can develop our final formula remember we need a formula that tells us what BC is in terms of things we know like the focal length of the lens the size of the aperture and the distance to our object P the key to this formula is to notice that there are similar triangles lurking in the diagram to figure this out let's simplify our diagram to show only the two triangles we care about notice the triangle ABC is similar to the triangle AED that means that the length BC divided by the length de is equal to the length a B divided by AE let's call this equation one we know what de is that's the radius of the aperture but we don't know a B or AE I'm also going to add two new points F and G this gives us two new right triangles a bf and a eg which are also similar this means that a B divided by a e equals FA divided by a G but FA is just the difference between I and I prime and a G is the distance I so we can rewrite this as a B divided by AE equals i prime minus I divided by I and now let's substitute this back into equation 1 which gives us BC divided by de equals I prime minus I divided by AI finally we just solved for BC which gives us our answer BC equals de times I prime minus I divided by I aha we have an equation for the circle of confusion if you wanted to you could also substitute I and I prime from the simple ends equation to write out the radius of the circle of confusion in terms of F o and O prime I'll leave that for you to work out in this final exercise