# If two opposite sides of a rectangle increase in length, how must the other two opposite sides...

## Question:

If two opposite sides of a rectangle increase in length, how must the other two opposite sides change if the area of the rectangle is to remain constant?

## Rectangles-Definition, Area, and Properties :

In the two dimensional geometry, a rectangle is a closed geometrical shape with four sides such that angle at each of the four vertices is a right angle also each side is parallel to the opposite side and have the identical length.

A rectangle has two diagonals of the identical length and the diagonal length of the rectangle is calculated by utilizing the Pythagorean theorem-

$$D^{2} = l^{2}+w^{2}$$

here $$D$$ is the length of the rectangle

{eq}l {/eq} is the length of the rectangle and {eq}w {/eq} is the width of the rectangle

#### Area of the Rectangle

$$A = l \times w$$

where A is the area of the rectangle in square units

Let the length and width of the rectangle be {eq}l {/eq} and {eq}w. {/eq}

$$A = l \times w --------------(1)$$

Also given that if two opposite sides of a rectangle increase in length.

Let the length of the rectangle is increased n times.

$$l' = n l$$

For the same area -

$$A = (l')(w')$$

$$A = (nl) (w')$$

from the equation(1)-

$$l( w) = n l (w')$$

$$w' = \frac{w}{n}$$

So the width of the rectangle is {eq}\displaystyle \frac{1}{n} {/eq} times the old width for the area of the rectangle is to remain constant. 