Pythagorean theorem: Interactive puzzle / Etudes // Mathematical ... - Free Printable

Let’s solve this step by step.

We are looking at a geometric figure made of several colored shapes arranged around a central right triangle (marked with the black square, which means it’s a 90° angle).

This is actually a visual proof of the Pythagorean Theorem — but we don’t need to know that name to solve it. We just need to notice:

- There are three squares built on the sides of a right triangle.
- One square is on the left (yellow + red + green parts) → let’s call its side length “a”
- One square is on the bottom (blue + purple part) → let’s call its side length “b”
- One big tilted square is on the hypotenuse (the slanted side) → let’s call its side length “c”

In the Pythagorean Theorem:
> a² + b² = c²

But here’s the trick — the colors show us how the areas relate.

Look closely:

The yellow square and the blue square together have the same total area as the big gray square? Not exactly — because they’re cut into pieces.

Actually, this diagram shows that if you take the two smaller squares (on the legs of the triangle), their combined area equals the area of the large square on the hypotenuse.

Wait — but in the image, the yellow shape isn’t a full square — it’s part of one. Let me re-express.

Actually, looking again — the entire left block (red + yellow + green) forms a square. The bottom block (purple + blue) forms another square. And the big gray one is the third square.

And since they’re attached to the sides of a right triangle, the sum of the areas of the two smaller squares equals the area of the largest square.

So if we assume the small square on the bottom has side length 1 unit (for simplicity), then its area is 1² = 1.

What about the left square? It looks bigger. But wait — maybe we can count units?

Actually, there’s no grid or numbers given. So perhaps the question is implied: what relationship holds between the areas?

But the user didn’t ask a specific question — they just uploaded an image.

Wait — maybe I missed something. Let me reread the instructions.

The user said: “Solve the problem accurately.” But there’s no explicit question in the image.

Hmm. This is tricky.

Perhaps the intended problem is: “What theorem does this illustrate?” or “What is the relationship between the areas?”

Given that, and since this is a classic dissection proof of the Pythagorean Theorem, the answer should be:

The sum of the areas of the two smaller squares equals the area of the largest square.

But to give a numerical answer — maybe we’re supposed to assume side lengths?

Looking at the diagram again — the small square at the bottom (blue + purple) appears to be half the size of the left square? Not necessarily.

Alternatively — perhaps the black square indicates the right angle, and the triangles around it are congruent or something.

Wait — here’s a better approach.

In many such diagrams, the colored regions are rearranged to show that the area of the two small squares fills the big square.

Specifically:

- The red triangle and the light blue triangle are congruent.
- The green triangle and the purple triangle are congruent.
- The yellow quadrilateral and the rest fill up.

Actually, this is a known proof called the “Bride’s Chair” or similar.

But without a specific question, I think the most reasonable assumption is that the task is to recognize that:

> Area of left square + Area of bottom square = Area of big gray square

Since no numbers are given, maybe the answer is simply stating that relationship.

But the instruction says “solve the problem”, implying there is a numeric or definite answer.

Wait — perhaps the small black square is 1x1, and we’re to find areas based on that?

Let’s try assigning values.

Assume the small black square (at the corner) has side length 1. Then the bottom square (blue + purple) has side length equal to... well, from the diagram, the bottom square seems to have side length equal to the height of the small triangle.

Actually, let’s define:

Let the vertical leg of the right triangle be ‘a’, horizontal leg be ‘b’, hypotenuse ‘c’.

Then:

- Left square has area a²
- Bottom square has area b²
- Big square has area c²

And by Pythagoras: a² + b² = c²

Again, same thing.

But maybe in this diagram, the areas are shown to be equal through rearrangement.

Looking at the colors:

- Red triangle moves to where light blue is?
- Green triangle moves to where purple is?
- Yellow stays?

Actually, yes — this is a standard proof where the two small squares are dissected and reassembled into the large square.

Therefore, the key point is: the total area of the two smaller squares equals the area of the large square.

Since no specific question is asked, but the user expects a final answer, I’ll go with the fundamental relationship.

Final Answer:
The sum of the areas of the two smaller squares equals the area of the largest square.