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Topic: X and Y are interior points (Read 848 times) |
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tony123
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X and Y are interior points
« on: Nov 5th, 2007, 2:28pm » |
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X and Y are interior points of the square ABCD such that <XAY= <XCY= 45°. Determine the length XY in terms of the lengths BX and DY.
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| « Last Edit: Nov 6th, 2007, 1:23am by tony123 » |
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Aryabhatta
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Re: X and Y are interior points
« Reply #1 on: Nov 6th, 2007, 1:14pm » |
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I get sqrt(|BX|2 + |DY|2)
I will post the approach I used later.
Note to tony, if you give a figure, it will be clearer, as there are two ways to label the segment XY and could lead to different answers depending on the labelling.
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Aryabhatta
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As promised. Please pardon the paint-job.
These square problems always make me want to reflect them!
Assume BX is the green line and DY is the red line (see top left square)
Tile up the square as in the image to create a BIGGER square BDBD.
We can show that XY is equal to RG (angle RAG is 45 and AR = AY, AG = AX)
Consider the center of that bigger square, say O. We can show that the angle between the red (OR) and green (OG) lines at the point is 90 (as the sum of four of them is 360, considering the four angles at O, which are equal to each other)
Thus we have a right triangle with BX and DY as the sides and XY as the hypotenuse.
Sorry, cannot hide the figure. Towr's idea to zip it.
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| « Last Edit: Nov 7th, 2007, 12:16pm by Aryabhatta » |
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towr
wu::riddles Moderator
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Re: X and Y are interior points
« Reply #3 on: Nov 7th, 2007, 11:13am » |
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on Nov 7th, 2007, 10:43am, Aryabhatta wrote:| Sorry, cannot hide the figure. |
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If you zip it before attaching, it wouldn't get shown.
Quote:| We can show that the angle between the red (OR) and green (OG) lines at the point is 90 (as the sum of four of them is 360, considering the four angles at O) |
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It doesn't looks like it in the drawing (but maybe it's not that accurate). And to be honest I don't really see why it ought to be. The angles at all corners of two crossing lines always add to 360 degrees, so that really doesn't help me.
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| « Last Edit: Nov 7th, 2007, 11:22am by towr » |
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Wikipedia, Google, Mathworld, Integer sequence DB
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Aryabhatta
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Re: X and Y are interior points
« Reply #4 on: Nov 7th, 2007, 11:32am » |
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on Nov 7th, 2007, 11:13am, towr wrote:
It doesn't looks like it in the drawing (but maybe it's not that accurate). And to be honest I don't really see why it ought to be. The angles at all corners of two crossing lines always add to 360 degrees, so that really doesn't help me. |
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Yes. Sorry, I forgot to mention that the four angles are equal to each other... triangles OGR and OGY are congruent.
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| « Last Edit: Nov 7th, 2007, 11:36am by Aryabhatta » |
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Joe Fendel
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Re: X and Y are interior points
« Reply #5 on: Nov 7th, 2007, 4:33pm » |
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Very clever, Aryabhatta! 
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Barukh
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Re: X and Y are interior points
« Reply #6 on: Nov 10th, 2007, 4:58am » |
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on Nov 7th, 2007, 4:33pm, Joe Fendel wrote:| Very clever, Aryabhatta! |
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Indeed! 
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square_45_soln.zip