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If someone walks up to you[br]on the street and says,

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all right, I have a[br]challenge for you.

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I want to construct a triangle[br]that has sides of length 2.

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So sides of length-- let me[br]write this a little bit neater.

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Sides of length 2, 2, and 5.

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Can you do this?

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Well, let's try to do it.

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And we'll start with the longest[br]side, the side of length 5.

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So the side of length 5.

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That's that side[br]right over there.

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And now, let's try to draw[br]the sides of length 2.

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Every side on a[br]triangle, obviously,

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connects with every other side.

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So that's one side of length 2.

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And then this is another[br]side of lengths 2.

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Another side of length 2.

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And you might say, fine, these[br]aren't touching right now,

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these two points.

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In order to make a triangle,[br]we have to touch them.

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So let me move them[br]closer to each other.

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But we have to remember, we[br]have to keep these side lengths

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the same.

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And we have to keep touching[br]the side of length 5

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at its endpoint.

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So we could try to move them in.

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We could try to move them in,[br]but what's going to happen?

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Well, you could rotate[br]them all the way down

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and they're still not going[br]to touch because 2 plus 2

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is still not equal to 5.

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They rotate all the[br]way down, they're

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still going to be 1 apart.

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So you cannot construct[br]this triangle.

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You cannot construct[br]this triangle.

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And I think you're noticing[br]a property of triangles.

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The longest side cannot[br]be longer than the sum

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of the other two sides.

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Here, the sum of the[br]other two sides is 4.

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2 plus 2 is 4.

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And the other side is longer.

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And even if the other[br]side was exactly

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equal to the sum of[br]the other two sides,

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you're going to have[br]a degenerate triangle.

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Let me draw that.

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So this would be side,[br]say, 2, 2, and 4.

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So let's draw the[br]side of length 4.

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Side of length 4.

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Side of length 4.

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Let me draw it a[br]little bit shorter.

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So that's your side of length 4.

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And then, in order to make the[br]two sides of length 2 touch,

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in order to make them touch,[br]you have to rotate them

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all the way inward You[br]have to rotate them

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all the way inward so that[br]both this angle and this angle

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essentially have to[br]become 0 degrees.

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And so your resulting triangle,[br]if you rotate this one

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all the way in and you[br]rotate this all the way in,

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the points will actually touch.

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But this triangle will[br]have no area anymore.

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This will become a[br]degenerate triangle.

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And it really looks more[br]like a line segment.

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So let me write that down.

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This is a degenerate.

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In order for you to draw[br]a non-degenerate triangle,

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the sum of the[br]other two sides have

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to be longer than[br]the longest side.

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So for example, you could[br]definitely draw a triangle

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with sides of[br]length 3, 3, and 5.

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So if that's the side of[br]length 5, and then this--

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if you were to rotate[br]all the way in,

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those two points would-- let me[br]draw this a little bit neater.

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So let's say that's[br]where they connect.

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And we know that[br]we could do that,

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because if you think about it,[br]if you were to keep rotating

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these, they're going to pass[br]each other at some point.

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They're going to[br]have to overlap.

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If you tried to make[br]a degenerate triangle,

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these points wouldn't touch.

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They'd actually overlap by[br]one unit right over here.

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So you could rotate them[br]out and actually form

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a non-degenerate triangle.

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So this one, you[br]absolutely could.

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And then there's another[br]interesting question,

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is this the only triangle[br]that you could construct

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that has sides of[br]length 3, 3, and 5?

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Well, you can't[br]change this length.

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So you can't change that[br]point and that point.

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And then, you can't[br]change these two lengths.

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So the only place[br]where they will

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be able to touch each other is[br]going to be right over there.

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So this right over here[br]is the only triangle

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that meets those constraints.

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You could rotate it[br]and whatever else.

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But if you rotate this, it's[br]still the same triangle.

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This is the only triangle[br]that has sides of length 3, 3,

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and 5.

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You can't change[br]any of the angles

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somehow to get a[br]different triangle.

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