Problem 2: Squares are drawn as shown (see link) on the long sides of a shaded rectangle. ?

Question

A diagonal of the resulting figure cuts off a 5-12-13 triangle from each square, as shown. What is the area of the shaded rectangle?

http://artpad.art.com/gallery/?k5xnd01621h8 (click "skip to end")

Answer 1

Suppose you draw this on coordinate paper (x,y), where the bottom edge of the rectangle is at 0,0.  The width of the rectangle is 24+whr, where whr is the width of the shaded strip.  So the other end of the bottom is at (24+wlr).

Now, the tricky part.  Consider only the BOTTOM HALF of the rectangle (in term of the y-axis).  The height of this half-rectangle is lhr+5.   Draw the two diagionals of the initial rectange.  They intersect at the point ([24+wlr]/2, 5+lhr), which is at the half-way point along the x-axis and at the height of the half-rectangle.

Now draw in the inside sides of the squares.   Starting from the top, go down lhr distance on the inside square sides.   They meet the two 5/12/13 triangles at this y-position.   So we will have a small rectangle along the top of the half-rectangle centered, of width wlr and length lhr.  

By laws of similar triangles, the portion of the diagionals inside the small rectangle will have the proportions of 5/12/13.   This arrangement suggests a trial&error solution.

(1)  Assume a value of lhr.

(2) Based on the 5/12/13 ratio, find the non-hypotenuse side of the triangle with height lhr.  This will be wlr/2.

(3) Check to see if the dimensions 12+wlr=2(5+lhr)

If not, you don't have a square, and you will have to try another guess.  

(4) If you do, the strip area is wlr x 2(5+lhr)

Sorry I couldn't come up with a closed form equation.   I am sure that nottheja worked hard to come up with his solution, but I don't think his statement that the squares are 12 units on a side is correct.

Answer 2

the shaded rectangle is 12 by 4.8 for an area of 57.6

how, you might ask...

The problem tells you that squares are drawn on the long sides of the rectangle. since the base of each side figure is 12 (from the 5-12-13 triangle), that means the height of the figure is also 12

slope of diagonal is 5/12 (from the legs of the 5-12-13 triangles)

for the entire figure, slope = 12 / (24 + w), where w is the width of the shaded rectangle

12/(24 + w) = 5/12

cross-multiplying, 144 = 5(24 + w) = 120 + 5w

24 = 5w

w = 24/5 = 4.8

therefore, the shaded rectangle is 4.8 by 12 = 57.6 square units

Edit:  cattbarf... note the problem statement--

Problem 2: SQUARES are drawn as shown (see link) on the long sides of a shaded rectangle. ?

Since squares by definition have sides of equal length, the base of 12 means that the height also has to be 12.  Without this information, there is no way to come up with a unique solution.