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Visual Pythagorean Theorem Proof: Visual Pythagorean Theorem Proof - some basic geometry required
相關課程
- We've already seen several videos where we, one, learned
- what the Pythagorean theorem is and how it can be applied
- to find the sides of right triangles, and even to find
- the distances between points on the Cartesian plane, so
- this is the Pythagorean theorem.
- And just as a bit of review, it tells us that if I have a
- right triangle, and if c is the side opposite the right
- angle, right?
- That's my right angle. c is that side.
- It's the longest side.
- We call that the hypotenuse.
- Then the square of the other sides, the sum of the squares
- of the other sides, will be equal to c squared.
- So if this side is a and this side is b, the Pythagorean
- theorem tells us that a squared plus b squared is
- going to be equal to c squared.
- And we did several problems where we used this to solve
- different triangles.
- But what I want to show you in this video is really a
- visual-- you can view it as a visual proof of the
- Pythagorean theorem to see why this actually makes sense.
- And to understand that, I'm going to draw a little bit of
- a diagram here.
- So let me draw a little right triangle here.
- So let's say that that is my right triangle.
- And let's say that this is a, this is b,
- that is my right angle.
- Then that right there, this right here, that length right
- there is c.
- Now, what I'm going to do is draw this exact same triangle,
- but I'm going to rotate it a little bit.
- You're going to see what I'm talking about.
- This right here is the exact same triangle that I just
- drew, except I rotated it.
- So this is a, this is b, and now this is c.
- This is the exact-- you could do this, you know.
- You could take a ruler out and actually do it.
- Pick an a, b, and c, and you can actually rotate this
- triangle around to the right like this.
- You could just rotate it like that, and you would get this
- triangle, just like that.
- Now, I'm going to rotate it again.
- And I'm just going to imagine copy and pasting it.
- And then I'll get this triangle, just like that.
- And once again, this is a, this is b, and this is c.
- These triangles are all the exact same triangle, the exact
- same measure.
- Now let me rotate this again, and I'll get this triangle.
- I will get this triangle right here.
- And once again, this is a.
- This length right here is a.
- This length right here is b.
- And this is the hypotenuse of this right angle, this is c.
- Now what do we have here?
- We have all of these right angles, and inside, I have
- this other square that is inscribed
- into this right angle.
- And I can prove to you, if you like, that these are also
- right angles.
- And you know, why don't I just show it to you.
- You may or may not already be familiar that the angles in a
- triangle add up to 180.
- So if you call this angle x, you call this angle y, we know
- that x plus y plus 90 degrees have to be equal to 180.
- So x plus y plus 90 have to be equal to 180.
- Or you could say that if you subtract 90 from both sides of
- this equation, x plus y are going to be equal to 90.
- I just subtracted 90 from both sides.
- This disappears.
- 180 minus 90 is 90.
- And this x and y, they're all over.
- If this angle is x right here, so is this angle because this
- is the exact same triangle.
- So is this angle.
- So is this angle.
- If this angle is y, so is that angle, so is that angle, and
- so is that angle.
- Now, you may or may not already know that if you have
- three angles that form a line-- so let me draw it.
- If you have three angles that form a line like this, so
- let's say we have angle a, angle b-- let me draw b a
- little bit different-- and angle c, just like that, and
- together they kind of form you could imagine a semi-circle or
- they go from one line all the way back, that a, b plus c are
- going to be equal to 180.
- a plus b plus c is going to be equal to 180 degrees.
- They complete a 180-degree arc.
- Now, if you look at, let's say, this point right here on
- our diagram, you can see that angle x plus this angle right
- here, which I'll call angle z, x plus z plus y is going to be
- equal to 180 degrees.
- They form a line.
- So x plus z plus y, or maybe I should write it as x plus y
- plus z is equal to 180.
- Now, we already know x plus y is equal to 90, so this is
- equal to 90.
- So 90 plus z is equal to 180.
- z minus-- well, both sides of this equation minus 90
- degrees, you get 180 minus 90.
- I just subtract 90 from both sides, so z
- is equal to 90 degrees.
- So these are all 90-degree angles.
- Their sides are all the same length of c, so this is a
- square in the middle of this diagram.
- Now, let's think a little bit about what the area
- of the square is.
- And there's two ways we can come up with the
- area of that square.
- So let me draw something here.
- So the first easy way to think of the area of the square,
- well, it's a square.
- This length is c, that length is c.
- To figure out the area, I just multiply one side
- by the other side.
- It's c times c, or c squared.
- So if I want the area of that right there, it is c
- squared, c times c.
- Area is equal to c squared.
- But how else could we figure out that area?
- Well, we could say that would be the area of this larger
- square, of this whole square over here minus the area of
- these triangles.
- That also would give us the area of that inside square.
- So what is the area of this outside, bigger square?
- Well, each of its sides are a plus b.
- And so you take a plus b.
- If you want the area, you could say, a plus b times a
- plus b, or a plus b squared, right?
- That's the area of the outer square, but we still have to
- get the area of our inner square.
- Area of small square.
- We still have to subtract out the area of these triangles.
- And what are the areas of these triangles?
- Well, we have to subtract out.
- The area of each of them is 1/2 times base times height,
- or base times height, ab, 1/2 ab.
- That's the area of each of these triangles.
- Now we have 1, 2, 3, 4 of these triangles.
- So it's 4 times 1/2 ab.
- This is another way of getting the area of this inside
- smaller square.
- Now let's just do a little bit of algebra-- we're pretty
- familiar with this-- to essentially simplify the
- right-hand side of this equation right here.
- So a plus b times times a plus b, a times a is a squared,
- plus 2 ab, plus b squared.
- And then we have minus-- what do we have right here?
- Minus 2ab, right?
- 4 times 1/2 is 2, so minus 2 ab.
- Well, these cancel out.
- 2ab minus 2ab, we're just left with a squared and b squared.
- So the area of this inside circle, which is c squared,
- it's also equal to a squared plus b squared.
- And I've just to shown you a visual proof for the
- Pythagorean theorem, because remember, a and b were the
- smaller sides of a right triangle having c as its
- hypotenuse.
- And from there, we constructed this bigger diagram.
- Hopefully, you found that fun.
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