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        riddles >> medium >> X and Y are interior points
        
(Message started by: tony123 on Nov 5^th, 2007, 2:28pm)

Title: X and Y are interior points
Post by tony123 on Nov 5^th, 2007, 2:28pm

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.

Title: Re: X and Y are interior points
Post by Aryabhatta on Nov 6^th, 2007, 1:14pm

I get [hide] sqrt(|BX|² + |DY|²) [/hide]

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.

Title: Re: X and Y are interior points
Post by Aryabhatta on Nov 7^th, 2007, 10:43am

As promised. Please pardon the paint-job.

[hide]
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.

[/hide]

~~Sorry, cannot hide the figure.~~ Towr's idea to zip it.

Title: Re: X and Y are interior points
Post by towr on Nov 7^th, 2007, 11:13am

on 11/07/07 at 10:43:49, Aryabhatta wrote:

Sorry, cannot hide the figure.

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)

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.

Title: Re: X and Y are interior points
Post by Aryabhatta on Nov 7^th, 2007, 11:32am

on 11/07/07 at 11:13:57, towr wrote:

Yes. Sorry, I forgot to mention that the four angles are equal to each other... triangles OGR and OGY are congruent.

Title: Re: X and Y are interior points
Post by Joe Fendel on Nov 7^th, 2007, 4:33pm

Very clever, Aryabhatta! ;D

Title: Re: X and Y are interior points
Post by Barukh on Nov 10^th, 2007, 4:58am

on 11/07/07 at 16:33:27, Joe Fendel wrote:

Very clever, Aryabhatta! ;D

Indeed! :D