Let's write down the equations of the angle bisectors of the two regions determined by the lines $4 x-3 y=7$ and $12 x+5 y=7$.

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If the normal vectors of two lines have equal length, then their sum or difference gives exactly the normal vectors of the angle bisectors.

$$
\begin{array}{rll}
e: 4 x-3 y=7 & \mathbf{n}_{e}=(4 ;-3) & \left|\mathbf{n}_{e}\right|=5 \\
f: 12 x+5 y=7 & \mathbf{n}_{f}=(12 ; 5) & \left|\mathbf{n}_{f}\right|=13
\end{array}
$$

Scale $\mathbf{n}_{f}$ by a factor of $\frac{5}{13}$ to make it the same length as $\left|\mathbf{n}_{e}\right|$; with this, one of the angle bisectors has the equation:

$$
e+\frac{5}{13} \cdot f:\left(4+\frac{60}{13}\right) x+\left(-3+\frac{25}{13}\right) y=7+\frac{35}{13}
$$

After rearranging, we get the equation $8 x-y=9$.

The equation of the other angle bisector is:

$$
e-\frac{5}{13} \cdot f:\left(4-\frac{60}{13}\right) x+\left(-3-\frac{25}{13}\right) y=7-\frac{35}{13}
$$

After rearranging: $x+8 y=7$.

We know that the two angle bisectors are perpendicular to each other, which can also be read from their normal vectors.