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# Object image height and distance relationship

Video transcript

In the last video, I showed
that a bunch of triangles are similar to each
other to come up with the relationship
between the focal length, the distance of an object
from the convex lens, and the distance of the image
of an object from that convex lens. And I realized that there was
one extra low-hanging fruit based on all of the
geometry that I had done, another relationship that
might come in useful. And that's the relationship
between the size of the object-- or maybe
since it's an arrow here, we can call it the
height of the object, and the height of the image. And we really set up
everything already. We already figured out that
this triangle over here is similar to this
triangle down here. And we figured out
in the last video that this triangle over here
is similar to this triangle over here. And since these
two are similar, we could say that A is to
B-- we did this over here. I'll rewrite it. A is to B as-- and both
of those are the sides opposite the right angles of
these two similar triangles. So that's going to be the same
thing as the ratio of the sides opposite this yellow
angle right over here. So in this triangle over here,
since we started with A first, it's this height
right over here. Now what is this
height right over here? This is the height
of the object. So this is the
height of the object is to-- Now what is
this opposite side of this yellow angle
right over here? Well, this is the
height of the image. Or we know from the
last video the distance of the object to the
distance of the image is the same thing as A to B. So this is going to be
the same thing as this. So the ratio of the
distances is also the same thing as the
ratio of their heights. So let me write it this way. So the ratio of the distance
from the object to the lens, to the distance from
the image to the lens, is the same as the
ratio of the height of the object to the
height of an image, or to the image of that object. So I just wanted to do that
little low-hanging fruit there, since we set up all of
the mechanics already. Anyway, hopefully you
found that useful.