Question

select the correct line.
which line is perpendicular to \\( \overleftrightarrow{ab} \\)?

Explanation:

To solve this, we first need to find the slope of \(\overrightarrow{AB}\). Let's assume points \(A\) and \(B\) have coordinates. From the graph, if \(A\) is at \((-6, 0)\) and \(B\) is at \((6, 3)\) (estimating from the grid), the slope \(m_{AB}=\frac{y_2 - y_1}{x_2 - x_1}=\frac{3 - 0}{6 - (-6)}=\frac{3}{12}=\frac{1}{4}\). A line perpendicular to \(\overrightarrow{AB}\) will have a slope that is the negative reciprocal, so \(m_{\perp}=-4\). Now, looking at the lines: Line 1 and Line 2 have steep slopes. Let's check their slopes. If Line 1 goes through, say, \((-1, 8)\) and \((0, -4)\), slope is \(\frac{-4 - 8}{0 - (-1)}=-12\) (not -4). Line 2: if it goes through \((1, -4)\) and \((0, 0)\), slope is \(\frac{0 - (-4)}{0 - 1}=-4\). Wait, maybe better to recall that perpendicular lines have slopes that multiply to -1. If \(\overrightarrow{AB}\) has a small positive slope (shallow), the perpendicular should have a steep negative slope. Line 1 and Line 2 are steep. Let's re - evaluate. Suppose \(\overrightarrow{AB}\) has slope \(m\), then perpendicular slope is \(-\frac{1}{m}\) (wait, no: if \(m=\frac{1}{4}\), negative reciprocal is -4). So a line with slope -4. Looking at the graph, Line 1 or Line 2? Wait, maybe the original \(\overrightarrow{AB}\) is the shallow line (Line 3 or the red/purple lines). Wait, the problem is to find which line is perpendicular to \(\overrightarrow{AB}\). Let's assume \(\overrightarrow{AB}\) is the line with points \(A\) (left) and \(B\) (right, higher). So slope is positive and small. Then perpendicular slope is negative and large (steep). So between Line 1 and Line 2, which has slope -4? Let's take two points on Line 2: say, (0,0) and (1, -4). Slope is \(\frac{-4 - 0}{1 - 0}=-4\). If \(\overrightarrow{AB}\) has slope \(\frac{1}{4}\), then \(\frac{1}{4}\times(-4)=-1\), so they are perpendicular. So the correct line is Line 2 (or Line 1? Wait, maybe I mixed up. Wait, maybe \(\overrightarrow{AB}\) has slope \(\frac{3 - 1}{6 - (-6)}=\frac{2}{12}=\frac{1}{6}\), then perpendicular slope is -6. But looking at the graph, Line 1 and Line 2 are the steep ones. Let's check the answer: usually, in such problems, the perpendicular line to a shallow positive - sloped line is a steep negative - sloped line. So among Line 1 and Line 2, let's see their directions. Line 1 goes from top - left to bottom - right (negative slope, steep), Line 2 also top - left to bottom - right? Wait, no, Line 2 in the graph (blue) goes from bottom - left to top - right? Wait, no, the user's graph: Line 1 (green) is steep left - top to right - bottom, Line 2 (blue) is steep left - bottom to right - top? Wait, maybe I misread. Wait, the key is that perpendicular lines have slopes that are negative reciprocals. If \(\overrightarrow{AB}\) has a positive slope (shallow), perpendicular has negative slope (steep). So the steep negative - sloped line is Line 1? Wait, maybe the correct answer is Line 1 or Line 2. Wait, maybe the intended answer is Line 2 (or Line 1). But based on typical problems, if \(\overrightarrow{AB}\) is the shallow line, the perpendicular is the steep line with negative reciprocal slope. So the correct line is Line 2 (assuming the slope calculations).

Answer:

Line 2 (or the specific line labeled as having the steep negative slope that is the negative reciprocal of \(\overrightarrow{AB}\)'s slope)