Two similar rectangles. If the small one has area 4 what is the area of the large rectangle?

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## Solution

The large rectangle has area 9.

We will find the solution for a general angle α. Calling the large width W, its height will be Wtan(α). Now, from parallel lines and the similarity of the rectangles, we can fill in the other angles as shown.

In the 2α right triangle, the hypotenuse w=Wtan(α)/sin(2α)=W/(2cos^{2}(α)). Hence W/w=2cos^{2}(α). Because of similarity, the proportion large rectangle area : small rectangle area is the square of that: 4cos^{4}(α).

## Poem

An orange rectangle is hiding

From figures who want

To know him

He is shy, and lonely feeling

Wants to have friends and outside going

Only with love and not discriminating

From circles, squares or triangles threatening

He is not complaining

Just so many feelings overwhelming