3.1: Squares in Rectangles

Rectangle $ABCD$ is not a square. Rectangle $ABEF$ is a square.

Suppose segment $AF$ were 5 units long and segment $FD$ were 2 units long. How long would segment $AD$ be?

Suppose segment $BC$ were 10 units long and segment $BE$ were 6 units long. How long would segment $EC$ be?

Suppose segment $AF$ were 12 units long and segment $FD$ were 5 units long. How long would segment $FE$ be?

Suppose segment $AD$ were 9 units long and segment $AB$ were 5 units long. How long would segment $FD$ be?


Rectangle $JKXW$ has been decomposed into squares.
Segment $JK$ is 33 units long and segment $JW$ is 75 units long. Find the areas of all of the squares in the diagram.

Rectangle $ABCD$ is 16 units by 5 units.

In the diagram, draw a line segment that decomposes $ABCD$ into two regions: a square that is the largest possible and a new rectangle.

Draw another line segment that decomposes the new rectangle into two regions: a square that is the largest possible and another new rectangle.

Keep going until rectangle $ABCD$ is entirely decomposed into squares.
 List the side lengths of all the squares in your diagram.


The diagram shows that rectangle $VWYZ$ has been decomposed into three squares. What could the side lengths of this rectangle be?

How many different side lengths can you find for rectangle $VWYZ$?

What are some rules for possible side lengths of rectangle $VWYZ$?