Divide convex polygon into 4 equal areas

In summary, to divide a convex polygon into 4 equal areas, the diagonal method can be used by drawing two diagonals from opposite vertices. It is possible to divide any convex polygon into 4 equal areas using this method, but it may not work for non-convex polygons or polygons with holes. A convex polygon is defined as having all interior angles less than 180 degrees and all vertices pointing outwards. Other methods such as the perpendicular bisector, centroid, and parallel lines methods can also be used, but the diagonal method is the most commonly used and easiest. A concave polygon cannot be divided into 4 equal areas using the diagonal method, and a different method would need to be used.
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Homework Statement



Show that it is possible to cut any convex polygon into 4 pieces of equal areas by using two cuts perpendicular to each other.

Homework Equations


None, it's just a proof I found on the back of my book. The relevant chapter is Continuity, the maximum principle, and intermediate value theorem for real analysis.


The Attempt at a Solution



I have no idea! Can someone tell me how to approach this please?

Thanks!
 
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  • #2
Anyone know how to do this?
 
  • #3
What about considering the centroid of the convex polygon?
 

Related to Divide convex polygon into 4 equal areas

1. How do you divide a convex polygon into 4 equal areas?

To divide a convex polygon into 4 equal areas, you can use the diagonal method. First, draw a diagonal from any vertex to the opposite vertex. Then, draw another diagonal from the remaining two vertices. This will divide the polygon into four equal triangles, resulting in four equal areas.

2. Is it possible to divide any convex polygon into 4 equal areas?

Yes, it is possible to divide any convex polygon into 4 equal areas using the diagonal method. However, this method may not work for non-convex polygons or polygons with holes.

3. What is a convex polygon?

A convex polygon is a polygon where all interior angles are less than 180 degrees and all of its vertices point outwards. In other words, any line segment connecting two points inside the polygon will be entirely contained within the polygon.

4. Are there any other methods to divide a convex polygon into 4 equal areas?

Yes, there are other methods such as the perpendicular bisector method, the centroid method, and the parallel lines method. However, the diagonal method is the most commonly used and the easiest to apply.

5. Can a concave polygon be divided into 4 equal areas?

No, a concave polygon cannot be divided into 4 equal areas using the diagonal method. This method only works for convex polygons. In order to divide a concave polygon into 4 equal areas, a different method such as the parallel lines method would need to be used.

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