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Jimmy Snyder
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That L shaped thing reminded me of a puzzle. How do you draw a line that bisects an L-shaped plane figure? In this case, bisect means cut into two pieces of the same area.
That is precisely what I mean.BicycleTree said:L-shaped thing? What exactly do you mean--any rectangle with a rectangular hole cut out of a corner, or something more specific?
jimmysnyder said:Let me add that there are infinitely many lines that bisect the L shaped figure. Consider any boundary point of the L shape that is on the outer edge of the larger rectangle. Consider a line that goes through that point and is parallel to the line that it is on. Then let that line sweep across the figure like a radar sweep. At some point it will sweep half the area.
The method given as a solution to the problem is the only one I know of that is constructable with straight edge and compass.
I have my doubts about this. Take a line that makes a 20 degree angle with one of the sides of the L shaped figure and pass that line through the figure until it passes half of the area. This line is the bisector through all of its points, but it is not constuctable.NateTG said:Actually, it's not that difficult to construct a bisector given an aribitrary point that the bisector must go through, but it can be tedious.
jimmysnyder said:I have my doubts about this. Take a line that makes a 20 degree angle with one of the sides of the L shaped figure and pass that line through the figure until it passes half of the area. This line is the bisector through all of its points, but it is not constuctable.
jimmysnyder said:But there is only one line passing through a given point on the boundary that bisects the figure. Since my 20 degree line is that line for at least two points, your tedious but not difficult program seems to me to be impossible.
Good point. Of those two lines, one of them behaves as I said, it intersects the boundary at two other points and is the only bisector that passes through those points. The other line however passes through the inside and outside corners of the L and at the inside corner there are two bisectors, so this line does not behave as I said it would.NateTG said:Well, actually, there may be more than one line that does so since the figure is not convex. (For example, on an L formed by cutting a 3x3 square out of a 5x5 square, there are 2 perpendicular lines that go through the 'crook' of the L that both bisect the L's area.)
The issue with 20 degree angles is not that they don't exist, but that they are not constructable. I don't need to construct the 20 degree line (I can't), I merely need to note that it exists. Ergo, I do not 'provide a line that is at a 20 degree angle' and you have nothing to translate.NateTG said:If you provide a line that is at a 20 degree angle from the horizontal, or any line at all, then it can be translated so that it bisects the figure using a compass and straightedge in finitely many steps.
jimmysnyder said:However, the existence of such a line has nothing to do with the main point I was trying to make. What I am saying is that you cannot construct a bisector through an arbitrary point in an arbitrary L shaped figure as you had indicated. I am not 100% sure that I am correct, but I think the image of that 20 degree line and its intersection with at least one 'outside' boundary point is compelling.
You are leaving out an important part of the assertion, namely that the 20 degree angle line is the ONLY bisector that goes through that point. Since you can't construct the 20 degree angle line, you lose the only chance you might have had to construct a bisector through that point.NateTG said:You're asserting that it is impossible to construct a bisector through some (plane figure) and a given point because it is impossible to construct a bisector through the same (plane figure) at a 20 degree angle to the horizontal - or if you prefer, some other arbitrary line.
jimmysnyder said:You are leaving out an important part of the assertion, namely that the 20 degree angle line is the ONLY bisector that goes through that point. Since you can't construct the 20 degree angle line, you lose the only chance you might have had to construct a bisector through that point.
Obviously, such an L shaped figure is itself not constructable.BicycleTree said:what if the L-shape is originally designed so that the angle through the centers of the two rectangles is 20 degrees? You wouldn't say that the method didn't work in that case.
Nate is right about a great many things and he has correctly pointed out an error in something I wrote. I only have issue with one thing. He implied that there is a constructable bisector passing through each point of each L shaped figure. I claim that my program displays a counter-example to this. Please carry it out.BicycleTree said:Nate's right.
Why are you drawing the bisector before you draw the point that you want to draw the bisector through? That doesn't make sense. Nate isn't claiming that you have the power to draw a 20 degree bisector, he's claiming that once you have drawn an L and a point, you can draw a bisector of the L that passes through the point. So you draw the L and the point first, and these must follow your stipulation that they are constructable. Then you draw the bisector. If you succeed at this and the bisector makes a 20 degree angle with anything, then perhaps it was your point that was not constructable, not the bisector.jimmysnyder said:You are leaving out an important part of the assertion, namely that the 20 degree angle line is the ONLY bisector that goes through that point. Since you can't construct the 20 degree angle line, you lose the only chance you might have had to construct a bisector through that point.
Please carry out this program:
1. Draw, freehand, an L shaped figure, not too symmetric, not too assymetric. Roughly is good enough, no need to be too finicky about it.
2. Draw, freehand, a line through the figure so that it bisects it, and so that it makes a 20 degree angle with one of the sides. Roughly is good enough, no need to be too finicky about it.
3. Mark, in red, a point where the bisector intersects with the boundary, but not on the two 'inside' lines.
4. Note that there is only one bisector that passes through that red point. Why? Consider a radar sweep with the red point as the center. It will only get one chance to sweep out half the area of the figure.
5. Get back to me. At that time I will explain the implications of this program.
BicycleTree said:Nate, your recent post wasn't clear to me. Maybe jimmy can understand what you mean but in places I can't (for example, "the line segment from the exterior corner to five units in"). Could you post a diagram or something?
In fact, on my way home from work I realized this too.NateTG said:I was trying to come up with an easy example. I just realized that there is an even easier one:
A rectangle is an L.
It's sort of beating a dead horse now, but for future reference please undertand this: I only need to show that a bad point exists, I do not need to show that the bad point is constructable. Nate said he could construct a bisector through any point. If he had said that he could construct a bisector through any constructable point, my intuition would fail me.BicycleTree said:So you draw the L and the point first, and these must follow your stipulation that they are constructable.
jimmysnyder said:My intuition still tells me that there are unconstructable bisectors on the L, but I can't prove it and besides Nate has already implied that he has a program (tedious, but not difficult) that constructs them. Can you share it with us? Until you do, I consider this an open question and I intend to pursue my intuition by looking for a counter-example.
Given the length, isn't it very easy to construct a rectangle that has that length as the length of one of its sides? What's the polygon for?Given a polygon, and a length it is possible to construct a rectangle that has that length as the length of one of its sides in a finite number of steps.
What does this mean?Using a reversible construction for the geometric mean (there are several options) allows for the creation of a rectangle with the specified dimension as one of its sides. For each triangle.
"Areas on either side of a vertex"?Pick some vertex to start with, and measure the areas on either sides of the vertex, if the areas on either side of the region are equal, we're done and that is the bisector.
Otherwise, sweep through the vertices until which side has more area flips.
I assume that you mean the following:NateTG said:Given a polygon, and a length it is possible to construct a rectangle that has that length as the length of one of its sides in a finite number of steps.
Hurkyl said:I wonder if it would not be easier to simply write down equations defining the bisector in terms of the various parameters, and see if it's solvable with square roots.
Bisecting an L-shaped plane figure means to divide it into two equal parts by drawing a line that cuts the figure in half.
To bisect an L-shaped plane figure, you can use a straightedge or ruler to draw a line that connects the opposite corners of the figure. This line will divide the figure into two equal parts.
Bisecting an L-shaped plane figure is important in geometry because it allows us to find the midpoint of the figure, which can be used to calculate other measurements and angles within the figure.
Yes, there are multiple ways to bisect an L-shaped plane figure. One way is to draw a line connecting the opposite corners, but you can also bisect the figure by drawing a line through the midpoint of one side and the opposite corner.
Any shape with two or more sides can be bisected, including triangles, squares, rectangles, and more complex shapes. The method of bisecting may vary depending on the shape, but the concept remains the same.