We are given a triangle with angles $A$, $B$, and $C$, and side lengths $a$, $b$, and $c$. We are given the following information: $C = 124.4^\circ$, $B = 25.6^\circ$, and $c = 39.2$ cm. We need to find the side length $a$.

1. Problem Description

We are given a triangle with angles $A$, $B$, and $C$, and side lengths $a$, $b$, and $c$. We are given the following information: $C = 124.4^\circ$, $B = 25.6^\circ$, and $c = 39.2$ cm. We need to find the side length $a$.

2. Solution Steps

First, we need to find angle $A$. The sum of the angles in a triangle is $180^\circ$, so:

$A + B + C = 180^\circ$

$A + 25.6^\circ + 124.4^\circ = 180^\circ$

$A + 150^\circ = 180^\circ$

$A = 180^\circ - 150^\circ$

$A = 30^\circ$

Now, we can use the Law of Sines to find the side length $a$:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

We have $A=30^\circ$, $C=124.4^\circ$, $c = 39.2$ cm. We want to find $a$, so we can use:

$\frac{a}{\sin A} = \frac{c}{\sin C}$

$\frac{a}{\sin 30^\circ} = \frac{39.2}{\sin 124.4^\circ}$

$a = \frac{39.2 \cdot \sin 30^\circ}{\sin 124.4^\circ}$

We know that $\sin 30^\circ = 0.5$. We can calculate $\sin 124.4^\circ \approx 0.8258$.

$a = \frac{39.2 \cdot 0.5}{0.8258}$

$a = \frac{19.6}{0.8258}$

$a \approx 23.73$ cm

3. Final Answer

$a \approx 23.73$ cm