# Object image and focal distance relationship

### Focal length - Wikipedia

More content like this is available at hidden-facts.info The focal length of an optical system is a measure of how strongly the system converges or When a lens is used to form an image of some object, the distance from the object to the lens u, the distance from the lens to the image v, and the. Image and Focal Distance Relationship, High School Physics.

So what we can do is-- and the whole strategy-- I'm going to keep looking for similar triangles, and then try to see if I can find relationship, or ratios, that relate these three things to each other. So let me find some similar triangles.

So the best thing I could think of to do is let me redraw this triangle over here. Let me just flip it over. Let me just draw the same triangle on the right-hand side of this diagram. So if I were to draw the same triangle, it would look like this. And let me just be clear, this is this triangle right over here. I just flipped it over.

And so if we want to make sure we're keeping track of the same sides, if this length right here is d sub 0, or d naught sometimes we could call it, or d0, whatever you want to call it, then this length up here is also going to be d0. And the reason why I want to do that is because now we can do something interesting. We can relate this triangle up here to this triangle down here. And actually, we can see that they're going to be similar. And then we can get some ratios of sides.

And then what we're going to do is try to show that this triangle over here is similar to this triangle over here, get a couple of more ratios. And then we might be able to relate all of these things. So the first thing we have to prove to ourselves is that those triangles really are similar. So the first thing to realize, this angle right here is definitely the same thing as that angle right over there.

They're sometimes called opposite angles or vertical angles. They're on the opposite side of lines that are intersecting. So they're going to be equal.

Now, the next thing-- and this comes out of the fact that both of these lines-- this line is parallel to that line right over there.

And I guess you could call it alternate interior angles, if you look at the angles game, or the parallel lines or the transversal of parallel lines from geometry. We know that this angle, since they're alternate interior angles, this angle is going to be the same value as this angle.

You could view this line right here as a transversal of two parallel lines. These are alternate interior angles, so they will be the same. Now, we can make that exact same argument for this angle and this angle. And so what we see is this triangle up here has the same three angles as this triangle down here. So these two triangles are similar. These are both-- Is really more of a review of geometry than optics.

These are similar triangles. Similar-- I don't have to write triangles. And because they're similar, the ratios of corresponding sides are going to be the same.

So d0 corresponds to this. They're both opposite this pink angle. They're both opposite that pink angle. So the ratio of d0 to d let me write this over here. So the ratio of d0. Let me write this a little bit neater. The ratio of d0 to d1. So this is the ratio of corresponding sides-- is going to be the same thing. And let me make some labels here. That's going to be the same thing as the ratio of this side right over here.

This side right over here, I'll call that A. It's opposite this magenta angle right over here. That's going to be the same thing as the ratio of that side to this side over here, to side B.

And once again, we can keep track of it because side B is opposite the magenta angle on this bottom triangle. So that's how we know that this side, it's corresponding side in the other similar triangle is that one. They're both opposite the magenta angles. We've been able to relate these two things to these kind of two arbitrarily lengths.

But we need to somehow connect those to the focal length.

### Object image and focal distance relationship (proof of formula) (video) | Khan Academy

And to connect them to a focal length, what we might want to do is relate A and B. A sits on the same triangle as the focal length right over here. So let's look at this triangle right over here.

Let me put in a better color. So let's look at this triangle right over here that I'm highlighting in green. This triangle in green. And let's look at that in comparison to this triangle that I'm also highlighting. This triangle that I'm also highlighting in green. Now, the first thing I want to show you is that these are also similar triangles. This angle right over here and this angle are going to be the same.

They are opposite angles of intersecting lines. And then, we can make a similar argument-- alternate interior angles. Well, there's a couple arguments we could make.

One, you can see that this is a right angle right over here. This is a right angle. If two angles of two triangles are the same, the third angle also has to be the same. So we could also say that this thing-- let me do this in another color because I don't want to be repetitive too much with the colors. We can say that this thing is going to be the same thing as this thing.

Or another way you could have said it, is you could have said, well, this line over here, which is kind of represented by the lens, or the lens-- the line that is parallel to the lens or right along the lens is parallel to kind of the object right over there.

And then you could make the same alternate interior argument there. But the other thing is just, look. I have two triangles. Two of the angles in those two triangles are the same, so the third angle has to be the same. Now, since all three angles are the same, these are also both similar triangles. So we can do a similar thing.

We can say A is to B. Remember, both A and B are opposite the degree side. They're both the hypotenuse of the similar triangle. So A is to B as-- we could say this base length right here.

- Converging Lenses - Object-Image Relations
- Object image height and distance relationship
- Focal length

And it got overwritten a little bit. But this base length right over here is f. That's our focal length. As f in this triangle is related to this length on this triangle. They are both opposite that white angle. So as f is to this length right over here. And we figured out in the last video that this triangle over here is similar to this triangle over here.

## Object image and focal distance relationship (proof of formula)

And since these two are similar, we could say that A is to B-- we did this over here. 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.