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Time wrote on 2010-12-29 21:52
I have to prove that the 6 small triangles of a centroid triangle have the same area...
[Image: http://upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Triangle.Centroid.svg/182px-Triangle.Centroid.svg.png]
This is created by finding the midpoint of one side, and connecting it to the opposite angle, now, I know that there are 6 triangles with the same area, but I dont know how to prove it....My teacher said something about it having to do with the formula for the area of a triangle...But I still dont get how to prove it.
:what:
Also, the lines all have a ratio of 2 to one, like, the midpoint to the other angle lines, the longer part is twice as long as the shorter part.
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psyal wrote on 2010-12-29 22:46
I'll give you a hint, as explaining the whole thing wouldn't let you do anything, plus it'd be hard to do not in person for me.
Each pair of triangles on each outside edge are equal area, due to having the same base length and height.
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Raj wrote on 2010-12-29 22:48
I want to answer (We're just studying that now in 8th grade over here).
But I'm not going to, I'm afraid of being wrong without graduating.
Though, I would say do use the type of triangle (the big overall one) and the lines of congruency.
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Time wrote on 2010-12-29 22:52
The way I see, each median divides the triangle into 2 parts of equal area, now, since this is true for each individual median, when you put all three into effect, and it creates 6 triangles, each one has the three medians dividing the area kinda, idk how to explain it, like, its because the dividing the are in half thing remains true for each line even if you add more lines.
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Skye wrote on 2010-12-29 22:54
I don't remember any fancy geometry terms, since I had the class two years ago. But hopefully this will help. ^^;
[Image: http://i262.photobucket.com/albums/ii107/LOVE__gaara_of_the_desert/proofcopy.png]
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Kueh wrote on 2010-12-30 00:15
Here's a hint.
Use ASA, and SAS similarity.
Then work in a circle.
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Kueh wrote on 2010-12-30 00:37
^ That solution doesn't work because you didn't prove that the areas are the same.
Here's mine:
Because any of the two triangles with congruent sides are the same area due to having the same base an height, the whole triangle can be shown as three groups of two triangles with the same areas.
Now the largest triangle can be shown as two triangles with each one containing one group, and one triangle from the last group, so that each new triangle is made up of a group with some area, and one triangle that is the same as the other, so that the area of each can be written like this.
Left Triangle Area = Right Triangle Area
Group 1 + Triangle from Group 3 = Group 2 + Triangle from Group 3
Because the triangles from group 3 are the same, they can be canceled.
Group 1 = Group 2
Which can be written as
2(Triangle from Group 1) = 2(Triangle from Group 2)
Triangle from Group 1 = Triangle from Group 2
Now you know that all the triangles in group 1 and group 2 are of the same area, and you can repeat the process to prove that triangles from group 1 and group 2 have the same area as triangle from group 3.
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Time wrote on 2010-12-31 02:48
I think I have a good idea of how to say this now, you guys have all kinda given me parts to put it all together, Thanks everyone!!!!!!