What rectangle is this?
The area of a rectangle is 2700 square units, and its diagonal is 75 units long. What are the length and width of the rectangle?
One solution (algebraic)
Let x = the length of the rectangle and y = the width. We want to find these.
Let A = the area (here, A = 2700 square units)
Let d = the diagonal (here, d = 75 units)
Then
x*y = A
(by the Pythagorean theorem)
Now we can substitute in 2700 for A and 75 for d, and we get
x = 60 units and y = 45 units.
Another solution (geometric)
A third solution.
This lesson was adapted from a Babylonian clay tablet almost 400 years old. It is described by Jöran Friberg in A Remarkable Collection of Babylonian Mathematical Texts, Springer 2007, on page 251.
The Babylonians did not use algebraic notation, so all problems were worked out with specific numbers. Also, drawn pictures were used very sparingly. Thus the actual problem and its solution looked more like this:
A diagonal of a rectangle is 10 and the area is 48. What are its length and width?
| 10 square makes 100 | d2 = x2 + y2 | |
| twice 48 makes 96 | 2xy | |
| 100 and 96 make 196 | x2 + 2xy + y2 = 196 | |
| root of 196 is 14 | x + y = 14 | |
| 100 less 96 makes 4 | x2 + y2 - 2xy = (x - y)2 = 4 | |
| root of 4 is 2 | x - y | |
| 14 and 2 make 16 | x + y + x - y = 2x | |
| half of 16 is 8 | it is the length | x |
| 14 less 2 make 12 | x + y - (x - y) = 2y | |
| half of 12 is 6 | it is the width | y |
Compare this computation to your algebraic solution and the geometric drawing to match the steps that led toward the solution given here.