Problem of Squares

[hemi]

I've been presented with this problem...

" If the smallest square has dimensions 1 x 1, what is the AREA of the rectangle ABCD. This problem is not to be solved using a ruler. "


The square then looks like:

Where I drew a line out is the smallest square, 1 x 1. Sorry about the bad drawing... In total there's 9 squares within the big square so I guess you could say 10 total?

So how does one go about solving this problem? Any solutions would be great.

 

[wjm11]

Where I drew a line out is the smallest square, 1 x 1. Sorry about the bad drawing... In total there's 9 squares within the big square so I guess you could say 10 total?

So how does one go about solving this problem? Any solutions would be great.

Hello, hemi,

Are you sure there wasn’t any other info given in the problem statement? There’s not really enough info in the picture you provided to arrive at the area of ABCD. Perhaps each rectangle is an exact multiple of the 1x1 square???

Likely you’re supposed to figure out the area of each rectangle, based on the 1x1 square. For example, maybe the rectangle just to the left of the square is 2x4 units (I can’t really tell), so its area would be 8.

Find all the areas, then add them together.

 

[soroban]

Hello, hemi!

I must assume that some information was omitted:
. . that the rectangle ABCD is divided into nine different squares.

If the smallest square has dimensions 1 x 1, what is the area of rectangle ABCD?

I'll draw the bottom half of the diagram.
I hope you can complete it and continue my reasoning.

      |               |                         |
  x+3 |               |   x                     |
      |               * - - - * - - - - - - - - *
      |     x+2     1 |///////|                 |
      * - - - - - * - *///////| x               |
      |          1|:::|///////|                 |
      |           * - * - - - *                 |
  x+2 |           | 1     x   |                 | 2x+1
      |           |           | x+1             |
      |           |           |                 |
      |           |           |                 |
      * - - - - - * - - - - - * - - - - - - - - *
           x+2         x+1           2x+1

\(\displaystyle \text{The 1x1 square is shaded :}\)
\(\displaystyle \text{The square to its right, shaded (///), is }x\text{-by-}x.\)

\(\displaystyle \text{The bottom has length: }(x+2)+(x+1)+(2x+1) \:=\:4x+4\)

\(\displaystyle \text{Continuing the diagram upward, the top is: }(x+7) + (x+11) \:=\: 2x+18\)

\(\displaystyle \text{Since }ABCD\text{ is a rectangle: }\:4x+4 \:=\:2x+18 \quad\Rightarrow\quad x \:=\:7\)
. . \(\displaystyle \text{Hence, the rectangle is: }\;32\text{ units wide.}\)


\(\displaystyle \text{We will find that the height is: }\x+7) + (x+3) + (x+2) \:=\:3x+12\)
. . \(\displaystyle \text{Hence, the rectangle is }\:33\text{ units high.}\)


\(\displaystyle \text{Therefore, the total area is: }\:32 \times 33 \:=\:1056\text{ units}^2.\)