A quick puzzler

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Famine

Famine

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Rule 12
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I was asked the other day to solve this for someone:

Draw a quadrilateral, using exactly one pair of parallel lines and two pairs of perpendicular lines.

I have one solution and some friends have managed a couple more which I think all fit the rules... Now I want to try GTP's geometry skills...
 


Two horizontal lines are the parallel lines.

Overlapped on the higher horizontal line is one of the components of one of the pairs of perp lines, the vertical line is the other component of the perpendicular line.
Same for the lower horizontal line, on the other side.
 
You have TWO pairs of parallel lines there - a horizontal pair and a vertical pair.
 
??

Edit: Hmmm... didn't use the second set of perpendicular lines...

That should be easy though.
 

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There are six lines in this picture - two different sets of perpendicular lines at the middle.
 

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It is - it's a complex quadrilateral. The two diagonal lines don't make up two sides of each triangle, but one side each of the quadrilateral.
 
The two sets of perpendicular lines make up each of the left and right sides of the quadrilateral and the parallel lines make up the top and bottom of the figure.

They may not be *perfectly* perpendicular or *perfectly* parallel, but they are. And the parallel lines are close to being perpendicular to the set of perpendicular lines on the left, but they are not. I would have drawn this better but its a 20 second job in MSPaint.
 

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VTGT07
The two sets of perpendicular lines make up each of the left and right sides of the quadrilateral and the parallel lines make up the top and bottom of the figure.

They may not be *perfectly* perpendicular or *perfectly* parallel, but they are. And the parallel lines are close to being perpendicular to the set of perpendicular lines on the left, but they are not. I would have drawn this better but its a 20 second job in MSPaint.
Click to expand...

But you didn't use the perpendicular lines.
 
Whats the V on the left there for? It serves no use in making the parallelogram.

You also have three sets of parallel lines. You can only have one.
 
The W shape seems to be useless; its lines aren't perpendicular to anything, though they could be parallel to themselves. You can have a max of only six lines, whereas you have seven.
The two horizontal lines could be your parallets, and the vertical line seems like it could be counted twice as crossing both horizontal lines, therefore you'd have two sets of perpendicular lines. If you took away the W you wouldn't have an enclosed shape.
Can lines be counted twice, or must you have six lines?

never mind, it was just like dan's first try.

EDIT: Danoff's is legitimate because it's a twisted square, and would have two sets of perps if it were still a square. They just crossed eachother when he "twisted" it.
 
The other part of the W isn't needed, I was just messing around with different line combinations. Where do you see the 3rd parallel?

Edit: Nevermind, I see its the W. I'll go make some touchups :D


untitled3sc.png
 
Hrm, can’t come up with anything other than Dan’s complex quadrilateral – we’re not allowed to overlay lines, right? And we’re not allowed to count two pairs of perpendicular lines that so happen to be parallel the “parallel lines”, right?

If that’s the case, then it would seem to me that it would be physically impossible to use these lines to make a simple quadrilateral, because you have a minimum of six lines that have to be used.
 
Radicools Mum
Well this is my effort
quad.jpg

I think it fits the rules.
Click to expand...

It doesn't look like you're using the perpendicular lines.

Sage
If that’s the case, then it would seem to me that it would be physically impossible to use these lines to make a simple quadrilateral, because you have a minimum of six lines that have to be used.
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:) Check this out.

There are only three lines here, but two sets of perpendicular and one sets of parallel.
 

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Yeah, but that’s what I was asking in my post above. ;) I wasn’t sure if you were allowed to “combine” lines like that.
 
danoff
It doesn't look like you're using the perpendicular lines.



:) Check this out.

There are only three lines here, but two sets of perpendicular and one sets of parallel.
Click to expand...

Perpendicular means two lines meeting at right angles. For the sake of this exercise you will have to assume I am a whiz at using paint and that those lines are at right angles.

Whilst your drawing follows the rules of 2 paralell and 2 perpendicular lines it doesnt make a quadrilateral.
 
Danoff was showing an example that you don't need to see 6 seperate lines in order to meet the requirements. He knows his stuff, and wasn't intending to make a quad out of that drawing.
 
Radicools Mum
Perpendicular means two lines meeting at right angles. For the sake of this exercise you will have to assume I am a whiz at using paint and that those lines are at right angles.
Click to expand...

I assumed that those lines were at right angles - but you aren't using them to make your quadrilateral.

Whilst your drawing follows the rules of 2 paralell and 2 perpendicular lines it doesnt make a quadrilateral.
Click to expand...

I was merely using to it point out to sage something he already knew - that fewer than 6 sides were necessary if the same line consituted part of a parallel pair and a perpendicular pair. I've already come up with a quadrilateral that satisfies the requirement.

Edit: Looks like keef beat me to it.
 
danoff
I assumed that those lines were at right angles - but you aren't using them to make your quadrilateral.



I was merely using to it point out to sage something he already knew - that fewer than 6 sides were necessary if the same line consituted part of a parallel pair and a perpendicular pair. I've already come up with a quadrilateral that satisfies the requirement.

Edit: Looks like keef beat me to it.
Click to expand...

I was just having a go at it, and used pretty much the same principles that everyone else had used. Lines going this way and that. I'd really like to see the solution so I can kick myself that I didnt think of it sooner.
 
There are a couple of solutions. Danoff created one earlier, and it makes perfect sense and is not just a bunch of lines going this way and that. Take a square and twist one end of it--you'll see two triangles, but, in fact, it is a square that has been twisted. Now everyone is trying to come up with a few other answers. You have to think outside the square.:lol: what a bad pun.
 
I'll post my solution then...

First off, I'd like you to imagine that I've drawn this shape using two pens attached to the middle of a ruler, "x" cm apart, in a single stroke. This means that if I maintain a consistent speed and unless I rotate the ruler about a point between the two pens the two resulting lines are implicitly parallel...

So... this is my lateral thinking solution... :D

famine-quad2.gif
 
Genius! That's what I call outside the box, it's way out there. But I think danoff's idea would make a mathematician/scientist happier from a geometric standpoint. Then again, yours is far more imaginative, so I like it.
 
Famine,

Very nice. Definitely outside-of-the-box thinking. I was trying to figure out how to use lines that are parallel on the surface of a sphere but I didn't manage anything.

However, I wonder if you've taken some liberties with the word "line" there. :)
 
A couple, perhaps... :D

But the definition of any line is a curve - in fact the definition of a straight line is a "perfectly straight curve"...

This was set for a colleague's daughter as maths homework.

For year 8 (12-13yo).
 

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