# Ratio of Areas in a Square

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

Given: a square with side length and diagonal . Place point on and let . Connect to divide the square into four areas. Given , calculate the ratio of areas. For example, let .

Contributed by: Abraham Gadalla (August 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The key here is that the height of equals the height of , so the ratio of the areas between those two triangles is .

Hence, .

Similarly, comparing areas, . Therefore |.

Substituting gives .

At this point, one can calculate the height of as .

Then and .

Also, .

After simplification, the ratio of the four areas is: .

## Permanent Citation