Ariel's work is not correct because of a mistake in the Pythagorean theorem's application. The Pythagorean theorem states that for a right triangle with side lengths \(a\), \(b\), and \(c\) (where \(c\) is the length of the hypotenuse), the relationship should be \(a^2 + b^2 = c^2\).
In this case, the side lengths are 13, 12, and 5. To determine if these form a right triangle, we should identify 13 as the longest side, which we would designate as the hypotenuse (c). Thus, we need to check if:
\[ 12^2 + 5^2 = 13^2. \]
Calculating this gives:
\[ 144 + 25 = 169. \] \[ 169 = 169. \]
Since both sides of the equation are equal, the sides 13, 12, and 5 do indeed form a right triangle. Therefore, Ariel's conclusion that these side lengths do not form a right triangle is incorrect. The proper verification of the Pythagorean theorem shows that the triangle is a right triangle.