find the length of the missing side of the right triangle.

Explanation:

Step1: Apply Pythagorean theorem

In a right - triangle, $a^{2}+b^{2}=c^{2}$, where $c$ is the hypotenuse. Here, if the missing side is $a$, and the other two sides are $b = 45.6$ and $c=49.4$. Then $a=\sqrt{c^{2}-b^{2}}$.

Step2: Substitute the values

$a=\sqrt{49.4^{2}-45.6^{2}}=\sqrt{(49.4 + 45.6)(49.4 - 45.6)}$ (using the difference - of - squares formula $x^{2}-y^{2}=(x + y)(x - y)$). First, calculate $49.4+45.6 = 95$ and $49.4 - 45.6=3.8$. Then $a=\sqrt{95\times3.8}=\sqrt{361}$.

Step3: Calculate the square root

$\sqrt{361}=19$.

Answer:

$19$