Arectangle measures by 3 cm by 4 cm. a rectangle similar to the first measures 12 cm by 16 cm. find the ratio of the areas of rectangles. how does that ratio compare with the ratio of the side lengths?

The side that measures 12 cm corresponds to the side that measures 3 cm.
The side that measures 16 cm corresponds to the side that measures 4 cm.
The ratio of the side lengths is (12 cm)/(3 cm) = (16 cm)/(4 cm) = 4

Now let's calculate the two areas.

For the 3 cm by 4 cm rectangle:
A = LW = 3 cm * 4 cm = 12 cm^2

For the 12 cm by 16 cm rectangle:
A = LW = 12 cm * 16 cm = 192 cm^2

The ratio of the areas is (192 cm^2)/(12 cm^2) = 16

The ratio of the areas is 16.
The ratio of side lengths is 4.
Notice that 16 = 4^2.
The ratio of the areas is the square of the ratio of the side lengths.