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Geometry Puzzle: Find the Area of the Square

Three squares: one big, one medium labeled/labelled nine and one small labeled/labelled one. The big square right bottom vertex is aligned with the medium square's base. The small square is on the top left of the medium square. The bottom left vertex of the big square and the top right vertex of the small square are the same. The top right vertex of the medium square touch the bottom side of the big square.

An orange square is tangent to a yellow and a green squares with areas 1 and 9, respectively. Its right bottom vertex is aligned with the green square’s base.

What is the area of the orange square?

Solution

The yellow square has area 1. Then, its side length is 1.

The green square has area 9. Then, its side length is 3.

Consider the small right(-angled) triangle with base 2 and height 1:

Three squares: one big, one medium and one small. The big square right bottom vertex is aligned with the medium square's base. The small square is on the top left of the medium square. The bottom left vertex of the big square and the top right vertex of the small square are the same. The top right vertex of the medium square touch the bottom side of the big square. The right side of the small square is labeled/labelled one. The top side of the medium square where the the small square is not over is labeled/labelled two. The right side of the medium square is labeled/labelled three.

Using the Pythagorean/Pythagoras’ theorem theorem, its hypotenuse is equal to

12+22\sqrt{1^2+2^2}
1+4\sqrt{1+4}
5\sqrt{5}

Now, consider the bigger right(-angled) triangle. It is similar to the small right(-angled) triangle with scale factor 3. Then, its base has length 6 and height 353\sqrt{5}.

Three squares: one big, one medium and one small. The big square right bottom vertex is aligned with the medium square's base. The small square is on the top left of the medium square. The bottom left vertex of the big square and the top right vertex of the small square are the same. The top right vertex of the medium square touch the bottom side of the big square. The right side of the small square is labeled/labelled one. The top side of the medium square where the the small square is not over is labeled/labelled two. They are the sides of a right/right-angled triangle which hypotenuse is labelled square root of five. A bigger right(-angled) triangle is below it, formed by the right side of the medium square labeled/labelled three, the line segment between the bottom right vertex of the medium square and the bottom right vertex of the big square labeled/labelled six, and the hypotenuse labelled three square root of five.

Then, the side length of the orange square is

5+35\sqrt{5}+3\sqrt{5}
454\sqrt{5}

So, the area of the orange square is

(45)2\left(4\sqrt{5}\right)^2
16×516 \times 5
8080

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