qwen2-math:1.5b

  clearly define the theorem and its proof step-by-step.

**Step 1: State the Theorem**

The Pythagorean Theorem states that 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. If we
denote the hypotenuse by $c$ and the other two sides by $a$ and $b$, then:

$$\begin{array}{}\text{(1)}& c^2=a^2+b^2\end{array}$$

**Step 2: Identify the Components of the Theorem**

- $c$: length of the hypotenuse
- $a$: length of one leg (one of the other two sides)
- $b$: length of the other leg

**Step 3: Explanation of the Formula**

The Pythagorean Theorem can be understood visually by considering a square constructed on each side of the right
triangle. Specifically:

- A square is constructed on each of the sides $a$, $b$, and $c$.
- The area of the square on the hypotenuse ($c^2$) equals the sum of the areas of squares on the other two
sides ($a^2+b^2$).

**Step 4: Proof by Geometric Interpretation**

To prove this geometrically, we can draw and manipulate shapes. Consider constructing a square with side length $(a+b)$. Then, inside this square, place four copies of the right triangle such that they form a square in the
center with side length $c$.

The area of the large square is:

$$\begin{array}{}\text{(2)}& (a+b{)}^2=a^2+2ab+b^2\end{array}$$

Within this large square, the four triangles each have an area of $\frac{1}{2}ab$. Since there are four such
triangles, their combined area is:

$$\begin{array}{}\text{(3)}& 4\left(\frac{1}{2}ab\right)=2ab\end{array}$$

The remaining portion inside the large square but outside the triangles forms a smaller square with side length $c$, and its area is therefore:

$$\begin{array}{}\text{(4)}& c^2\end{array}$$

Equating these areas, we get:

$$\begin{array}{}\text{(5)}& (a+b{)}^2=c^2+2ab\end{array}$$

Simplifying this, we have:

$$\begin{array}{}\text{(6)}& a^2+2ab+b^2=c^2+2ab\end{array}$$

By subtracting $2ab$ from both sides, we obtain:

$$\begin{array}{}\text{(7)}& a^2+b^2=c^2\end{array}$$

Thus, the Pythagorean Theorem is proven.

$$\begin{array}{}\text{(8)}& \boxed{{\displaystyle c^2=a^2+b^2}}\end{array}$$