# What is a long distance relationship definition math

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But before it got refracted, it traveled the distance from the object to the actual lens. Now, the distance from the image to the lens, that's this right over here. This is how far this parallel light ray had to travel. So this is the distance from the image to the lens.

And then we have the focal distance, the focal length. And that's just this distance right here. This right here is our focal length.

Or, we could view it on this side as well. This right here is also our focal length.

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

So I want to come up with some relationship. And to do that, I'm going to draw some triangles here. 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.

**Stages Of A Long-Distance Relationship**

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.

### Long-distance relationship - Wikipedia

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.

- Long-distance relationship

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. 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. Inverse Square Law The principle in physics that the effect of certain forces, such as light, sound, and gravity, on an object varies by the inverse square of the distance between the object and the source of the force. In physics, an inverse-square law is any physical law stating that a specified physical quantity or intensity is inversely proportional to the square of the distance from the source of that physical quantity.

The fundamental cause for this can be understood as geometric dilution corresponding to point-source radiation into three-dimensional space. One of the famous inverse square laws relates to the attraction of two masses. Two masses at a given distance place equal and opposite forces of attraction on one another. The magnitude of this force of attraction is given by: The graph of this equation is shown below.

More on Brightness and the inverse square law Damping Motion Damping is an influence within or upon an oscillatory system that has the effect of reducing, restricting or preventing its oscillations. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples include viscous drag in mechanical systems, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators.

Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems. Sine Wave Relationship The graphs of the sine and cosine functions are sinusoids of different phases. The sine wave or sinusoid is a mathematical curve that describes a smooth repetitive oscillation. It is named after the function sine, of which it is the graph.

Lab Activities and Resources What are Mathematical Relationships What is a mathematical relationship and what are the different types of mathematical relationships that apply to the laboratory exercises in the following activities. What is the relationship between how much a spring stretches and the force pulling on the spring? What is the relationship between the mass of a ball and its volume assuming a constant density?

What is the relationship between the intensity of a beam of light and the distance from a light source? What is a the relationship between how the distance travels and the time in travel for an accelerating object?

What is the relationship between how much light passes through a Polaroid filter and the angle the filter is rotated? What is the relationship between current, voltage when there is a constant resistance in an electric circuit. Radioactive Decay- - Problem: What is the relationship between the decay of radioactive material and the time allowed for the decay? Water Pressure - - Problem: What is the relationship between water pressure and depth of water? What is the relationship between the height a ball bounces and the number of times it has bounced?