Triangle inequality theorem
Triangle inqequality theorem Intuition behind the triangle inequality theorem
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- Let's draw ourselves a triangle.
- Let's say this (blue) side has lenght 6, this (pink) side has lenght 10 and this (green) side has lenght x
- And what I'm going to think about is how large or how small x can be
- How large or how small can this (green) side be.
- The first question is "How small can it get?"
- Well, if we want to make this small we would just have to look at this (green) angle here and make it smaller
- So let's try to make this angle as small as possible.
- So we have our side of lenght 10, and I'm going to make this angle really really small, approaching 0
- If that angle becomes 0 we end up with a degenerate triangle
- Essentially it becomes one-dimensional, we lose our 2-dimensionality
- But as we approach 0 this side starts to coincide or get closer and closer to the 10 side.
- And you could imagine the case where it actually coincides with it and you actually get the degenerate
- So if we want this point here (on the 10 side) to get as close as possible to that point over there (on the 6 side)
- -essentially minimizing your distance x-
- the closest way is that you make the angle all the way equal to 0
- Let me draw a progression. Now the angle is getting smaller, this (blue) side is lenght 6
- x is getting smaller
- so we keep making that angle smaller and smaller all the way until we get a degenerage triangle
- So you have the (pink) side of lenght 10. Now the angle that we care about is essentially 0
- This (blue) side is lenght 6
- And so what is the distance between this point and this point? That distance is lenght x
- So in the degenerate case, this (green) length right over here is x
- and we know that 6 + x is going to equal 10
- So in this degenerate case x will equal 4
- So if you want this to be a real triangle, at x = 4 you've gotten these points as close as possible
- It's degenerated into a line segment.
- If you want this to be a triangle x has to be greater than 4
- Now let's think about it the other way.
- How large can x be?
- To think about larger and larger x's we need to make this (green) angle bigger
- So let's try to do that.
- Let's draw the 10 side again. This is my 10 side
- and we make that angle bigger and bigger.
- So now let me take the 6 side and put it like that.
- So now our angle is getting bigger and bigger. It's approaching 180 degrees.
- At 180 degrees our triangle once again will be turned into a line segment, becoming a degenerate triangle.
- Let me draw the side of lenght x.
- So we are trying to maximize the distance between that point (on the 10 side) and that point (on the 6 side).
- This is the side of lenght x
- Let's go all the way to the degenerate case.
- At the degerate case, at 180 degrees,
- the side of lenght 6 forms a straight line with the side of lenght 10.
- And this is how you can get this point and that point as far appart as possible.
- In this situation, what is the distance between that point (on the 6 side) and that point (on the 10 side)?
- which is the distance that's going to be our x
- In this situation x is going to be 6 + 10 = 16
- If x is 16 we have a degenerate triangle.
- If we don't want a degenerate triangle, then x is going to have to be less than 16
- Now the whole principle that we are working on right over here is called the Triangle Inequality Theorem.
- It's a pretty basic idea.
- Any one side of a triangle has to be less than the sum of the other two sides.
- So the lenght of a side has to be less than the sum of the lenghts of the other two sides.
- If you're willing to deal with degenerate triangles,
- where you essentially form a line segment, lose all your dimensionality and turn to a 1-dimesional figure,
- then you could say "less than or equal"
- but we're just going to stick with non-degerate triangles.
- So the lenght of a side has to be less than the sum of the lenghts of the other two sides
- And just using this principle we could have come up with the same exact conclusion
- You could say "Well, look: x is one of the sides and it has to be less than 6 + 10 or x has to be less than 16"
- The exact same result we got by visualizing it like this
- If you want to say how "How small can x be?" you could say:
- "Well, 10 has to be less than 6 + x, the sum of the lenghts of the other two sides"
- If you subtract 6 from both sides right over here, you get 4 < x, or x > 4.
- So this is, in some level, kind of a basic idea.
- But it's something that you'll see definitely in geometry
- and then you'll go far into other types of mathematics and you'll see other versions
- of what's essentially this Triangle Inequality Theorem
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