About this video
Sal extends the similar-triangle work from the lens derivation to heights. The ratio of object distance to image distance equals the ratio of object height to image height, which gives a second relationship for magnification. He uses this ratio to decide whether an image is enlarged or reduced.
You'll be able to
- state the height ratio for an image formed by a convex lens.
- predict image size from the ratio of object and image distances.
- connect side ratios in similar triangles to magnification.
Key words
- Object height
- The perpendicular size of the object, measured across the principal axis.
- Image height
- The perpendicular size of the image formed by the lens.
- Magnification
- The ratio of image height to object height; values above 1 mean an enlarged image.
- Corresponding sides
- Sides in similar triangles that sit opposite equal angles, so their ratios match.
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.
Video © Khan Academy, used under CC BY-NC-SA. Watch on Khan Academy. Puzzicle is not affiliated with Khan Academy.