Proposed Problem

Click the figure below to see the complete problem 402 about Hexagon, Triangle, Centroid, Parallel, Congruence, Similarity.

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Complete Problem 402

Level: High School, SAT Prep, College geometry

## Thursday, December 10, 2009

### Problem 402. Hexagon, Triangle, Centroid, Parallel, Congruence, Similarity

Labels:
centroid,
congruence,
hexagon,
parallel,
similarity,
triangle

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My problems for this picture.

ReplyDeleteProff

1)Area hexagon ABCDEF/ Area hexagon GHJKLM = 9/4.

2) Area polygon ABCDE/ Area polygon GHJKL = 117/32.

2) Area polygon ABCD/ Area polygon GHJK = 9.

the affix of triangle ABC centroid G is

ReplyDeleteg=(a+b+c)/3

then

3(h-g)= b+c+d-a-b-c= d-a

3(k-l)= d+e+f-e+f+a= d-a

this means that GH=LK and GH // LK

.........

We see that

ReplyDeleteXH=(XB+XC+XD)/3

XG=(XB+XC+XA)/3

XK=(XD+XE+XF)/3

XL=(XE+XF+XA)/3

and HG=(XH-XG,YH-YG)=(XD-XA,YD-YA)/3

LK=(XK-XL,YK-YL)=(XD-XA,YD-YA)/3

etc.

And my question for this picture - prove

area hexagon ABCDEF/area hexagon GHJKLM = 9/4

Interesting about the areas, but the result about the formed hexagon having opposite sides equal & parallel can be generalized for 2n-gons as shown here: http://dynamicmathematicslearning.com/octagoncentroids.html

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