Question

use the image above to answer the following question. δabc is reflected onto δdef. what is the line of reflection?

Explanation:

Step1: Identify key points

First, find corresponding points of \( \triangle ABC \) and \( \triangle DEF \). Let's assume coordinates (estimating from grid): Let \( B = (-4,1) \), \( C = (-1,1) \); \( F = (1,1) \), \( E = (4,1) \). Also, \( A = (-4,6) \), \( D = (4,6) \).

Step2: Find midpoint

For a reflection, the line of reflection is the perpendicular bisector of the segment joining a point and its image. Take point \( B(-4,1) \) and its image \( F(1,1) \)? Wait, no, better take \( B(-4,1) \) and \( F(1,1) \)? Wait, maybe better \( B(-4,1) \) and \( F(1,1) \) – no, wait, actually, looking at the grid, the vertical line between \( x = -4 \) (B) and \( x = 4 \) (E? Wait, maybe I misread. Wait, \( \triangle ABC \) has B at \( x=-4 \), C at \( x=-1 \); \( \triangle DEF \) has F at \( x=1 \), E at \( x=4 \). Wait, the midpoint between \( x=-4 \) and \( x=4 \) (for A and D: A is at \( x=-4 \), D at \( x=4 \)): midpoint x-coordinate is \( \frac{-4 + 4}{2}=0 \)? No, wait, no. Wait, let's check the y-axis? Wait, no. Wait, the line of reflection should be the vertical line where the distance from each point to the line is equal. Let's take point B: let's say B is at \( (-4,1) \), F is at \( (1,1) \)? No, maybe I messed up coordinates. Wait, looking at the grid, the vertical line \( x = -1.5 \)? No, wait, actually, the line of reflection between \( \triangle ABC \) and \( \triangle DEF \) – let's check the x-coordinates. Let's assume:

Point A: let's say A is at \( (-3,6) \), D at \( (3,6) \). Point B: \( (-3,1) \), E at \( (3,1) \). Point C: \( (-1,1) \), F at \( (1,1) \). Wait, no, the grid lines: the vertical line between \( x=-3 \) and \( x=3 \) is \( x=0 \)? No, wait, the midpoint between \( x=-3 \) and \( x=3 \) is 0, but maybe the line is \( x = -1.5 \)? No, wait, actually, looking at the two triangles, \( \triangle ABC \) is on the left, \( \triangle DEF \) on the right. The line of reflection should be the vertical line that is the mirror. Let's check the distance from B to the line and F to the line. Suppose B is at \( x=-4 \), F at \( x=1 \) – no, maybe the correct line is \( x = -1.5 \)? Wait, no, maybe the line is \( x = -1 \)? No, wait, let's look at the grid again. Wait, the x-axis is labeled with -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6. The triangle \( \triangle ABC \): B is at (-4,1), C at (-1,1), A at (-4,6). \( \triangle DEF \): F at (1,1), E at (4,1), D at (4,6). So the midpoint between B(-4,1) and F(1,1) is \( (\frac{-4 + 1}{2}, 1) = (-1.5, 1) \)? No, that's not vertical. Wait, no, the vertical line: the x-coordinate of the midpoint between A(-4,6) and D(4,6) is \( \frac{-4 + 4}{2} = 0 \)? Wait, A is at (-4,6), D at (4,6) – so midpoint is (0,6), so the line of reflection is \( x = 0 \) (the y-axis)? Wait, but B is at (-4,1), E is at (4,1) – midpoint is (0,1), so yes, the line of reflection is \( x = 0 \) (the y-axis). Wait, but let's check: reflecting over y-axis (x=0) would map (x,y) to (-x,y). So A(-4,6) would map to (4,6) which is D. B(-4,1) maps to (4,1) which is E? Wait, no, E is at (4,1), but F is at (1,1). Wait, maybe my coordinate estimation is wrong. Let's re-estimate:

Looking at the grid, the first triangle (ABC): B is at x=-4, y=1; C is at x=-1, y=1; A is at x=-4, y=6. The second triangle (DEF): F is at x=1, y=1; E is at x=4, y=1; D is at x=4, y=6. So the vector from B to F is (1 - (-4), 1 - 1) = (5,0), but reflection should be symmetric. Wait, the midpoint between B(-4,1) and F(1,1) is (-1.5,1), between C(-1,1) and F(1,1) is (0,1), between A(-4,6) and D(4,6) is (0,6). Wait, that's inconsistent. Wait, maybe the line of reflection is \( x = -1.5 \)? No, that doesn't make sense. Wait, maybe the correct line is \( x = -1 \)? No. Wait, perhaps the line is \( x = 0 \) (y-axis). Let's check point C(-1,1): reflecting over y-axis gives (1,1), which is F. Yes! C(-1,1) reflects to F(1,1) over x=0. B(-4,1) reflects to (4,1), which is E. A(-4,6) reflects to (4,6), which is D. Yes! So the line of reflection is the y-axis, which is \( x = 0 \).

Answer:

The line of reflection is \( x = 0 \) (the y - axis).