Question

1.
reflection image \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\) across line \\( l \\)
step 1 pretend to fold the graph along line \\( l \\) to estimate where
points \\( a, b \\), and \\( c \\) will land.
step 2 draw a segment with endpoint \\( a \\) so that the segment is
perpendicular to line \\( l \\) and is bisected by line \\( l \\). label the
other endpoint of the segment \\( a ^ { prime } \\).
in this case, line \\( l \\) has a slope of \\( \frac { 1 } { 2 } \\) so segment \\( a a ^ { prime } \\)
must have an opposite - reciprocal slope of \\( - \frac { 2 } { 1 } \\).
step 3 repeat step 1 for vertices \\( b \\) and \\( c \\).
step 4 connect points \\( a ^ { prime }, b ^ { prime } \\), and \\( c ^ { prime } \\). \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\) is the reflected
image of \\( \triangle a b c \\). the image should match the estimate you
had in step 1.
refer to the bottom image
2.
line \\( l \\) has a slope of \\( - \frac { 2 } { 1 } \\), the segments must have an
opposite - reciprocal slope of \\( + \frac { 1 } { 2 } \\).
3.
line \\( l \\) has a slope of \\( 2 \\), the segments must have an
opposite - reciprocal of \\( - \frac { 1 } { 2 } \\).

Explanation:

Step1: Find the slope of line \( l \)

Looking at the grid, for line \( l \), if we consider two points on it (e.g., from \((0,0)\) to \((1,2)\)), the slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{2 - 0}{1 - 0}=2\).

Step2: Calculate the opposite - reciprocal slope

The formula for the opposite - reciprocal of a slope \( m \) is \(-\frac{1}{m}\). Given \( m = 2\), the opposite - reciprocal slope is \(-\frac{1}{2}\).

Answer:

Line \( l \) has a slope of \(2\), the segments must have an opposite - reciprocal of \(-\frac{1}{2}\).