Question:

Step 1: Identify the slope of the line parallel to the given reference line. The line $3x - y = 7$ can be rewritten as $y = 3x - 7$, which has slope $m = 3$. Any line parallel to this will also have slope $3$. Step 2: Write the equation of the line through point (1, 2) with slope 3. Using the point-slope form of a line: $$y - 2 = 3(x - 1)$$ $$y = 3x - 1$$ This is the line passing through $(1, 2)$ that is parallel to $3x - y = 7$. Step 3: Find the intersection of this line with $x + y + 5 = 0$. Substitute $y = 3x - 1$ into $x + y + 5 = 0$: $$x + (3x - 1) + 5 = 0$$ $$4x + 4 = 0$$ $$x = -1$$ Substituting back to find $y$: $$y = 3(-1) - 1 = -4$$ The intersection point is $(-1, -4)$. Step 4: Calculate the distance from (1, 2) to (-1, -4). Using the distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ $$d = \sqrt{(-1 - 1)^2 + (-4 - 2)^2}$$ $$d = \sqrt{(-2)^2 + (-6)^2}$$ $$d = \sqrt{4 + 36}$$ $$d = \sqrt{40}$$ $$d = \sqrt{4 \cdot 10}$$ $$d = 2\sqrt{10}$$ Step 5: Simplify and match with the given options. We can also write $\sqrt{40} = 2\sqrt{10}$. Comparing with the options provided, $\sqrt{40}$ matches **Option 3**. **Final Answer: The distance is $\sqrt{40}$ or equivalently $2\sqrt{10}$, which corresponds to Option 3.**