Question

which is the pythagorean theorem?
√(x - x)^2 - (y - y)^2
a² + b² = c²
(y - y)/(x - x)
a + b = c

Explanation:

Step1: Recall the Pythagorean theorem

The Pythagorean theorem states that in a right - angled triangle, the sum of the squares of the two legs ($a$ and $b$) is equal to the square of the hypotenuse ($c$).

Step2: Analyze each option

• For the option $\sqrt{(x - x)^{2}-(y - y)^{2}}$, this is not a valid formula for the Pythagorean theorem. In fact, $(x - x)^{2}-(y - y)^{2}=0$.
• For the option $a^{2}+b^{2}=c^{2}$, this is the standard form of the Pythagorean theorem where $a$ and $b$ are the lengths of the legs of a right - triangle and $c$ is the length of the hypotenuse.
• For the option $\frac{(y - y)}{(x - x)}$, this is an undefined expression (since $x - x = 0$ and $y - y=0$ in a non - trivial sense, and division by zero is undefined) and has nothing to do with the Pythagorean theorem.
• For the option $a + b=c$, in a triangle (by the triangle inequality $a + b>c$ for non - degenerate triangles), and this is not the Pythagorean relationship.

Answer:

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