1. The problem is to find the length of the third side of a right triangle where the two legs are given as $7$ and $\sqrt{51}$.\n2. We use the Pythagorean theorem for right triangles: $$c^2 = a^2 + b^2$$ where $c$ is the hypotenuse and $a$, $b$ are the legs.\n3. Substitute the given values: $$c^2 = 7^2 + (\sqrt{51})^2$$\n4. Calculate the squares: $$c^2 = 49 + 51$$\n5. Simplify the sum: $$c^2 = 100$$\n6. Take the square root of both sides: $$c = \sqrt{100}$$\n7. Simplify the square root: $$c = 10$$\n8. Therefore, the length of the third side (the hypotenuse) is $10$.