Question:

Step 1: Identify the vertex and the two angle bisectors. The triangle has vertex $A = (4, -1)$, and the two internal bisectors are $x - 1 = 0$ (vertical line) and $x - y - 1 = 0$. Step 2: Reflect the vertex $(4, -1)$ across the bisector $x = 1$. The reflection of $(4, -1)$ across $x = 1$ is $(-2+1\cdot2, -1) = (1-(4-1), -1) = (-2, -1)$. Step 3: Reflect the vertex $(4, -1)$ across the bisector $x - y - 1 = 0$. Using the reflection formula across $x - y - 1 = 0$: the reflection of $(4, -1)$ is found by $x' = y + 1 = -1+1 = 0$, $y' = x - 1 = 4-1 = 3$, giving $(0, 3)$. Step 4: The sides of the triangle opposite to vertex $A$ pass through the reflected points. The side through $(-2, -1)$ and $(0, 3)$: slope $= \frac{3-(-1)}{0-(-2)} = \frac{4}{2} = 2$, equation: $y - 3 = 2(x - 0) \Rightarrow 2x - y + 3 = 0$. Step 5: The other two sides pass through vertex $A = (4,-1)$ and the reflected points. Side through $(4,-1)$ and $(-2,-1)$: $y = -1 \Rightarrow y + 1 = 0$. Side through $(4,-1)$ and $(0,3)$: slope $= \frac{3-(-1)}{0-4} = -1$, equation: $y+1 = -1(x-4) \Rightarrow x + y - 3 = 0$. Step 6: The three sides of the triangle are $2x - y + 3 = 0$, $y + 1 = 0$, and $x + y - 3 = 0$. Matching with the given options, the correct sides are $2x + y - 7 = 0$ and $x - 2y - 6 = 0$.