Two simple geometry problems

Another short post for today: this time, we look at two geometry problems and how AI can solve them. I’m not doing a full benchmark now, like in the past, just an initial attempt.

The first problem comes from Catriona Agg: given the following image describing an orange rectangle and a black square, with the rectangle corner splitting the size of the square in equal parts and having a side of 4, what is the area of the rectangle?

The solution involves drawing an additional segment, such as in the following picture:

If the side of the square is 2a, then by similarity between the lower left right triangle and the triangle with the dashed side we have

\[\frac{4}{a} = \frac{x}{2a}\]

where \(x\) is the other side of the rectangle. From this, \(x = 8\), so the area is 32.

How did Gemini (using the app, Thinking mode of Gemini 3.0) solve this? Well, on first attempt it tried for nearly 15 minutes, and then it just deleted the chat. Hiding the failure means that there is no failure, right?

OpenAI’s ChatGPT didn’t fare much better here. It missed that the side of the square is split into equal parts and gave up, claiming that it needs more information.

Moving on, the second problem is similar, also from Catriona: given that the area of the large, white square is 25, what is the total shaded area?

This can also be solved by drawing the two diagonals:

If the side of the blue square is \(x\) and the other side of the rectangle is \(y\), we need to find \(x(x+y)\). But, we have two similar triangles, so

\[\frac{x\sqrt{2}}{5} = \frac{5\sqrt{2}}{x+y}\].

From this, the area is exactly 25.

Both Gemini and ChatGPT solved this problem, assuming that the shaded area is just a sheared version of the original square.

So, out of these problems, both AIs only solved half. Probably I would need to run a larger experiment, but this also means I have to solve all these problems myself. I really recommend looking through the old problems that Catriona shared.