Math Is Fun Forum

atreyyu

Member

Registered: 2008-08-01

Posts: 5

Line passing through a square

Straight line

passes through the center of a square

, whose area is

. Let

denote the shortest distances between line

and points

respectively. How to prove that

?

Last edited by atreyyu (2008-08-25 07:21:14)

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Line passing through a square

Look at the picture. It is sufficient to prove that

.
To show this, notice AO=DO=a half of the diagonal. The angle AOD is 90degs, so we end up with 2 congruent triangles, namely APO and ODQ. That means that OQ = AP, and by Pythagorean theorem,

Last edited by krassi_holmz (2008-08-25 09:00:12)

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mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Line passing through a square

Very nicely done!
I had a go at this but just ended up fumbling about with quadratic equations and trigonometry and didn't really get anywhere.

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Why did the vector cross the road?
It wanted to be normal.