Model of a pythagorean theorem

In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (legs). If the legs have lengths $a$ and $b$, and the hypotenuse has length $c$, the theorem is stated as:

$$a^2 + b^2 = c^2$$

A common way to model the Pythagorean theorem visually is by drawing squares on each side of a right-angled triangle. The area of each square represents the square of the length of that side.

• The area of the square on side $a$ is $A_a = a^2$.
• The area of the square on side $b$ is $A_b = b^2$.
• The area of the square on side $c$ (the hypotenuse) is $A_c = c^2$. The theorem states that the sum of the areas of the squares on the two shorter sides equals the area of the square on the hypotenuse:

$$A_a + A_b = A_c$$

Consider a right-angled triangle with legs of length $a = 3$ units and $b = 4$ units, and a hypotenuse of length $c = 5$ units.

• Area of the square on side $a$: $A_a = (3)^2 = 9$ square units.
• Area of the square on side $b$: $A_b = (4)^2 = 16$ square units.
• Area of the square on side $c$: $A_c = (5)^2 = 25$ square units.

$$A_a + A_b = 9 + 16 = 25$$

Since $A_c = 25$, we have $A_a + A_b = A_c$.
This visually demonstrates that the sum of the areas of the squares on the legs equals the area of the square on the hypotenuse.

The model of the Pythagorean theorem shows that for a right-angled triangle with sides $a$, $b$, and $c$ (hypotenuse), the relationship is $\boxed{a^2 + b^2 = c^2}$, which can be visualized as the sum of the areas of the squares on the legs equaling the area of the square on the hypotenuse.