Divide ABCDE into two parts with equal area

  • MHB
  • Thread starter Albert1
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In summary, the pentagon can be changed into a triangle with the same area by finding the midpoint of the line passing through the points A, B, and C and dividing the triangle into two equal parts.
  • #1
Albert1
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ABCDE is a pentagon,please construct a line (passing

through point A),and divide ABCDE into two parts with equal

area
 
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  • #2
Is this a general pentagon, or a particular pentagon?
 
  • #3
For a regular pentagon, just construct the midpoint of $CD$, call if $F$, and draw the segment $AF$. This splits the pentagon in two equal pieces.
 
  • #4
Prove It said:
Is this a general pentagon, or a particular pentagon?
A general pentagon(convex)
 
  • #5
Albert said:
ABCDE is a pentagon,please construct a line (passing

through point A),and divide ABCDE into two parts with equal

area
http://www.mathhelpboards.com/attachments/f28/998d1373594278-change-pentagon-into-triangle-equal-area-pentagon.jpg

​
Referring to Albert's beautiful solution to the problem in http://www.mathhelpboards.com/f28/change-pentagon-into-triangle-equal-area-5486/, if $M$ is the midpoint of $PQ$ then the line $AM$ will do the job, provided that $M$ lies between $C$ and $D$. I imagine that this must necessarily be the case, but I don't see how to prove it.
 
  • #6
In my opinion the best proof is "a proof without words"
so again I construct a diagram and let it explain the solution
View attachment 1033
and AG is what we need as written by Opalg
"if [FONT=MathJax_Math]M[/FONT] is the midpoint of [FONT=MathJax_Math]P[/FONT][FONT=MathJax_Math]Q[/FONT] then the line [FONT=MathJax_Math]A[/FONT][FONT=MathJax_Math]M[/FONT] will do the job"
 

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  • #7
Opalg said:
Referring to Albert's beautiful solution to the problem in http://www.mathhelpboards.com/f28/change-pentagon-into-triangle-equal-area-5486/, if $M$ is the midpoint of $PQ$ then the line $AM$ will do the job, provided that $M$ lies between $C$ and $D$. I imagine that this must necessarily be the case, but I don't see how to prove it.
The statement M lies between C and D is not always true
in fact M and C (or M and D)may coincide
May M also lie between D and E ?(if the length of CD is very small)
M may also lie between B and C.
(we may check this using various diagram)
 
Last edited:

Related to Divide ABCDE into two parts with equal area

1. How do I divide ABCDE into two parts with equal area?

To divide ABCDE into two parts with equal area, you will need to find the midpoint of the line connecting points A and E. This midpoint will divide the shape into two equal halves.

2. Is there a specific method for dividing ABCDE into two equal parts?

Yes, there are several methods for dividing ABCDE into two equal parts. One method is to use the midpoint of the line connecting points A and E, as mentioned in the previous answer. Another method is to draw a diagonal line from point E to point C, which will divide the shape into two equal triangles.

3. Can ABCDE be divided into two equal parts without using any measurements?

Yes, it is possible to divide ABCDE into two equal parts without using specific measurements. One way to do this is by drawing a line from point C to the midpoint of side AB, and another line from point D to the midpoint of side AE. These two lines will divide the shape into two equal triangles.

4. Will dividing ABCDE into two equal parts always result in equal area?

Yes, dividing ABCDE into two equal parts will always result in equal area. This is because the shape will be divided into two equal triangles or two equal halves, both of which have the same area.

5. Can I divide ABCDE into two parts with equal area using only a straightedge and compass?

Yes, it is possible to divide ABCDE into two parts with equal area using only a straightedge and compass. You can use the compass to find the midpoint of the line connecting points A and E and then use the straightedge to draw a line from this midpoint to point C or D (depending on the desired dividing line). This will divide the shape into two equal triangles.

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