Rhombus inscribed in rectangle

A

axrw

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Mar 18, 2007
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I have this problem that I can't seem to figure out. I know the answer (back of the book), but I don't know how they got there.

"A rhombus is inscribed in a rectangle that is w meters wide with a perimeter of 40m. Each vertex of the rhombus is a midpoint of a side of the rectangle. Express the area of the rhombus as a function of the rectangles width"

The answer being A(w) = 10w - (w^2)/2.

I figured I could get the side of a rhombus with a^2 + b^2 = c^2 in terms of w, but I don't know about the height. ...yeah, I'm so bad math. :?

None of it seems to be leading me to that function.

If anyone could shed any light on this I would very much appreciate it.
 
axrw, this is so simple that you'll kick yourself!
A rhombus' area = product of diagonals divided by 2

The diagonals are the width and length of the rectangle:
so the rhombus' area is half the rectangle's.

Rectangle is w wide, and 20-w high (since perimeter = 40).

So rhombus' area
= w(20-w) / 2
= (20w - w^2) / 2
= 10w - w^2/2

Did you bother to draw a rectangle with rhombus inside ?
 
*sighs* I think I'm going to cry. No I didn't draw one, and that just never occurred to me. I guess I should have paid more attention in geometry. :oops:

Thank you Denis, you wouldn't believe how much things like that bug me until I can see and understand why the answer is what it is.
 
axrw said:
Thank you Denis, you wouldn't believe how much things like that bug me until I can see and understand why the answer is what it is.
Click to expand...
Don't feel bad: that's the way I am too!
 
Hello, axrw!

Denis is correct (as usual), but diagonals need not be considered.
Code: 
      *-----------*-----------*
      |       *:::|:::*       |
      |   *:::::::|:::::::*   |
      *:-:-:-:-:-:+:-:-:-:-:-:*
      |   *:::::::|:::::::*   |
      |       *:::|:::*       |
      *-----------*-----------*

Eyeballing the diagram, it is obvious (isn't it?)
. . that the rhombus has one-half the area of the rectangle.
(We have four bisected rectangles.)

 
Upon viewing your ascii diagram (nice, by the way), it is clear.

It's also clear on problems involving geometry I need to draw them out. :oops:

Thanks Soroban.
 
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