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 ?

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- Aug 25th 2008, 11:31 AMatreyyuLine 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 ?

- Aug 25th 2008, 01:51 PMPlato
- Aug 25th 2008, 04:07 PMSoroban
Hello, atreyyu!

My approach is the same as Plato's . . .

Quote:

Straight line passes through the center of a square with area 1.

Let denote the shortest distances between line and vertices resp.

Prove: .

The line through center (½, ½) with slope m has the equation:

. .

**Formula**: The distance from point to line

. . . . . . . . is given by: .

We have: .

The distance is: .

For

For

For

For

Hence: .

. .

. .

- Aug 25th 2008, 05:20 PMPlato
Soraban, does your approach work if the line through the center is vertical?

- Aug 26th 2008, 03:02 AMatreyyu
- Aug 26th 2008, 04:29 AMSoroban
Hello, Plato!

Quote:

Does your approach work if the line through the center is vertical?

I believe it__does__work . . . since

Besides: .

. . Therefore: .