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Old 09-15-2020, 11:27 AM
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Default Geometry puzzle

What is x?

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  #2  
Old 09-15-2020, 11:52 AM
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Default Re: Geometry puzzle

what all do we know? are the circles true circles?

and like do we have enough information to say a number?

I'm not that mathy :tmgrin:
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Old 09-15-2020, 12:18 PM
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Default Re: Geometry puzzle

Quote:
Originally Posted by ceptimus View Post
What is x?
The 24th letter of the English alphabet, also widely used for an independent or general unknown.

Oh, you mean its value? 10 obviously.
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Old 09-15-2020, 01:06 PM
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Default Re: Geometry puzzle

Quote:
Originally Posted by beyelzu View Post
what all do we know? are the circles true circles?

and like do we have enough information to say a number?

I'm not that mathy :tmgrin:

It's a good question, and the simple answer is that we aren't given enough hard information for there to be a unique solution, or in other words, for the puzzle to be a puzzle. But then, this is true of all puzzles.

In order to derive a solvable puzzle you have to make a few assumptions, and I think they are these:

  • The "rectangle" is indeed a rectangle of sides 8 and x.
  • The two "circles" are indeed circles.
  • The line that appears to pass through the point where the two circles touch does indeed pass through that point.
  • That line is parallel with the side length 8 and perpendicular to the side length x.
It is somewhat surprising that that is enough to generate a genuine puzzle with a single solution, but I believe it is.
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Old 09-15-2020, 01:10 PM
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Default Re: Geometry puzzle

Quote:
Originally Posted by beyelzu View Post
what all do we know? are the circles true circles?

and like do we have enough information to say a number?

I'm not that mathy :tmgrin:
What we know:

(it's pretty much what you'd guess by looking at the diagram)

The rectangle has vertical and horizontal sides, and is 8 high. The puzzle is to find the width, x.

The two inscribed circles are true circles and touch each other tangentially (they just touch without overlapping). One circle tangentially touches one side and the top of the rectangle, the other circle tangentially touches the other side and the bottom.

A vertical line (parallel to the rectangle, side 8) has one end touching one circle, the other end touching the other circle. The line passes through the point where the circles touch each other but you'll find that last condition isn't necessary because the line always passes through that point if it meets the other conditions. The line is 6 long.

You're not told the size of the two circles, other than that they fit inside the rectangle as shown.

You've got enough information to find the answer, which is a constant.

One way of finding an approximate answer is to draw the diagram accurately and measure x. If you know how, you can use a CAD program to draw it, and get a very accurate approximation - but it's still just an approximation, and the real challenge is to use geometry or mathematics to find the answer.

JoeP's (numerical) answer is a reasonable guess... but


Clue
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Last edited by ceptimus; 09-16-2020 at 08:38 AM. Reason: I realized that the part I've now struck out was nonsense.
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Old 09-15-2020, 01:13 PM
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Default Re: Geometry puzzle

It's not wrong! Just not answering the question you have now clarified.

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  #7  
Old 09-15-2020, 01:16 PM
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Default Re: Geometry puzzle

One srs question: could this be solved with GCSE level maths?

It looks like the kind of thing that, when I did GCSE maths tuition (recent years but not this year), students would be horrified by but is actually p. straightforward. I say this not actually having tried to solve it yet.
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Old 09-15-2020, 01:16 PM
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Default Re: Geometry puzzle

:dddp:
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Old 09-15-2020, 01:22 PM
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Default Re: Geometry puzzle

The solution to these puzzles is almost always drawing the right lines.

I did a little doodle of the diagram and I've drawn so many lines on it that is just looks like graph paper with a couple circles drawn on it now.

I also have a bazillion variables like r and R to mark the radii of the circles, x,y,z... to mark different lengths.

So far I've gotten no where :chin:
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  #10  
Old 09-15-2020, 02:00 PM
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Default Re: Geometry puzzle

Quote:
Originally Posted by slimshady2357 View Post
The solution to these puzzles is almost always drawing the right lines.
Yes ... currently busybusy but I have a mental picture of the lines. And statements going round my head "opposite angles are equal", "the angle at the circumference is half the angle at the centre" ...
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Old 09-15-2020, 02:30 PM
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Default Re: Geometry puzzle



For each circle, O is the centre of the circle, C is the centre of the chord AB, D is the point on the circumference opposite.

eta: spoilerising things ppl might want to notice / work out

... if only we can work out those lengths.
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  #12  
Old 09-15-2020, 02:36 PM
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Default Re: Geometry puzzle



And now it's just trig ... I think
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Last edited by JoeP; 09-15-2020 at 02:38 PM. Reason: Spoilerific
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Old 09-15-2020, 02:45 PM
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Default Re: Geometry puzzle

OK since I haven't used the height 8 we're gonna need another couple of points.

E1, E2 = points on circumferences vertically below / above O1, O2.
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Old 09-15-2020, 02:49 PM
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Default Re: Geometry puzzle

Quote:
Originally Posted by JoeP View Post
Yes ... currently busybusy
That meeting finished early, and I didn't feel like preparing for the next one so I drew the above. Now busy again :deepsigh:
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Old 09-15-2020, 04:24 PM
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Default Re: Geometry puzzle

No further time for trig, but it has to be symmetrical therefore x=8.

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Old 09-15-2020, 04:25 PM
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Default Re: Geometry puzzle

Here's my solution:



Which is probably too complicated by half, but it gets an answer.
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Old 09-15-2020, 04:45 PM
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Default Re: Geometry puzzle

Looking at it again, I found a solution that doesn't involve sines or cosines, just Pythagoras. If people are interested I'll post in when I'm back at the computer (phone posting right now).
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Old 09-15-2020, 05:12 PM
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Default Re: Geometry puzzle

Pan wins the coveted golden turnip prize for the first correct answer!

:medal:
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Old 09-15-2020, 07:07 PM
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Default Re: Geometry puzzle

Quote:
Originally Posted by Pan Narrans View Post
Looking at it again, I found a solution that doesn't involve sines or cosines, just Pythagoras. If people are interested I'll post in when I'm back at the computer (phone posting right now).
I'm Interested!
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Old 09-15-2020, 07:24 PM
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Default Re: Geometry puzzle

I'm fairly confident that the centers of the circles and the point where they touch all lie on the diagonal of the rectangle, but it would take me a couple minutes to prove it.

Question: are the red line segments necessarily parallel? They look that way no matter what scale I draw them at, but I can't think of a reason they would be. Or really I guess my problem is I can't think of a reason they would be perpendicular to the line the centers lie on. AKA parallel to the line that is tangent to both circles, through the point where they meet.

circle1.png
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Old 09-15-2020, 07:38 PM
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Default Re: Geometry puzzle

Quote:
Originally Posted by Ensign Steve View Post
I'm fairly confident that the centers of the circles and the point where they touch all lie on the diagonal of the rectangle, but it would take me a couple minutes to prove it.

Question: are the red line segments necessarily parallel? They look that way no matter what scale I draw them at, but I can't think of a reason they would be. Or really I guess my problem is I can't think of a reason they would be perpendicular to the line the centers lie on. AKA parallel to the line that is tangent to both circles, through the point where they meet.

Attachment 11516
Well, reaching back many years, I seem to recall that for your first statement to be true, that would have to be a square.
But then the reason I went into tech support was so I wouldn't have to do all the Math.
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Old 09-15-2020, 07:40 PM
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Default Re: Geometry puzzle

:irwinnar:

So now for the simple solution:



Thanks for the puzzle, ceptimus. It was a fun one (and made me feel SMRT)
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  #23  
Old 09-15-2020, 07:48 PM
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Default Re: Geometry puzzle

Quote:
Originally Posted by Ensign Steve View Post
Question: are the red line segments necessarily parallel? They look that way no matter what scale I draw them at, but I can't think of a reason they would be. Or really I guess my problem is I can't think of a reason they would be perpendicular to the line the centers lie on. AKA parallel to the line that is tangent to both circles, through the point where they meet.

Attachment 11516
They are parallel. In the figure in my previous post it should be obvious (right?) that the angle ABC is equal to angle BA(all the way to the right), and that these are the half angles of the apices of isosceles triangle DBL and DAK. Therefore, the angle AK makes with the horizontal is equal to the angle BL makes with the horizontal, and thus they are parallel. Does that make sense?
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Old 10-14-2020, 03:03 PM
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Default Re: Geometry puzzle

Here's a 3:4:5 triangle drawn in Fusion 360, with a square drawn inside. The funny little symbols are the way you specify things like 'parallel', 'perpendicular', and 'equal length' in Fusion 360. The question is, what's the area of the square?

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Old 10-14-2020, 04:35 PM
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Default Re: Geometry puzzle

Guess 1: 42
Guess 2: something to do with the squaw on the hippopotamus.
Guess 3: 2.0, one-third the area of the triangle
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