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# 三角形面積證明 (英): 證明任何三角形的面積都是 1/2 底 x 高

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- now know how to find the areas of rectangles.
- What I want to do in this video, is think about how we can find the areas of triangles.
- so we are starting here with a right triangle. It has 90 degree angle. Triangle A B C
- so lets think about how we can find its area. Maybe we can construct a rectangle out of triangle ABC
- and if we can construct a reactangle out of it, maybe we can somehow
- find our area of part of that rectangle and the best way to construct our rectangle is to really duplicate
- ABC
- and then flip it over and put it right on top this. So just to verify that it will definitely be a rectangle
- so we know that this is 90 degrees right over here. Let's say that this is x degrees right over there
- we have x plus 90 plus this thing have to be equal to 180.
- so this thing and this thing have to add up to 90 so let's just call this 90-x.
- now lets flip this thing and rotate it around.
- so that it will look like this.
- so that you would have another triangle that would look just like this:
- where now this right angle on the flip version, is that right angle right over there
- this right angle right over here, this angle right over here, this x is now this angle right over there
- and this 90-x is now this angle right over there. now you can see x+90 -x that will give you a right
- angle
- and that will gives you a right angle, and you have four sides and four right angles you are definitely
- dealing with a rectangle.
- and this rectangle has two of our original triangles in it.
- So we can write that the area of triangle of ABC.
- So the area of triangle ABC, thats what the brackets mean. Area of triangle ABC
- is going to be equal to 1/2 times the area of our entire rectangle
- let me add another point here, let me call this: D, its going to be 1/2 the are of the rectangle.
- ABCD, and we know how to find the area of rectangle ABCD, its going to be equal to.
- Its going to be equal to the base of the rectangle. So this is going to be equal to 1/2 times this par.
- let me use this in a different color, the area of ABCD is equal to the base of the width of the rectangle
- so thats just the length of BC, I'm just putting this in parenthessis.
- BC is just the length of this length segment and I'm just putting it into parenthesis.
- so that we don't get the letters jumbled up. So its going to be this width or this base right here
- times the height of the rectangle. So times AD, sorry, times AB.
- so this base times this height is gives us the area of our entire rectangle
- And the area of our right triangle is half of that. So there we have it, 1/2 times times this base, times t
- this height.
- is the area of a right triangle. So in general if you ever have a right triangle and this is a right
- angle right over there. You need one right angle in order for it be a right triangle and this
- base has length B and this side over here has side h, so you know that the area
- the area is going to be equal to 1/2 times the base times the base of the triangle
- thats the base of the triangle bc.
- times the height of the triangle. so you can view it this way if you look at the actual letter points
- or just view these mesasures as the base times the height. 1/2 base times height
- and we only know if this works for a right triangle. now let's think about it for other types of triangle that aren't
- necessarily right triangles.
- So here I have a kind of a arbritrary triangle bc. and to approach figure out its area
- what I want to do is split this up into two right triangles. so what I'm going to do is drop
- a perpendicular from b. so i'm just going to literally, and if this was an actual structure you would just literally drop something straight down from here, and that line is
- going to be perpendicular to this base right over here. to ac let me call that point
- let me call that point d, and whats useful here is that now that we have constructed we turned that one
- triangle into two right triangles. So, we can say that the area
- of triangle ABC thats what we want to figure out is equal to the area of this character
- right here. So its equal to the area of triangle ABD.
- ABD plus the area of triangle, plus the area of this magenta triangle, of BCD.
- BCD
- And this is useful becuase we know how to find the area of right triangles
- and this is 90 degrees, this is also going to be 90 degrees.
- the area of ABD is 1/2 Base times Height.
- So its going to be 1/2 times AD times the height
- which is going to be the length of BD right over here, assuming that we can figure this out
- so this times that length. So BD, thats the area of the blue triangle.
- and now if and now I can find the lengrth of the magenta triangle.
- well once again its a right triangle its going to be 1/2, times the length of this base,
- right over here which is DC
- the length of segment DC times the length of BD again.
- Times the length of segment bd. Now you can factor out a 1/2

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