Question

given △abc and its reflection image △abc find the line of reflection. draw the line of reflection on the graph on the right. click to enlarge graph

Explanation:

Step1: Identify corresponding points

Let the coordinates of vertices of $\triangle ABC$ be $A(x_1,y_1)$, $B(x_2,y_2)$, $C(x_3,y_3)$ and of $\triangle A'B'C'$ be $A'(x_1',y_1')$, $B'(x_2',y_2')$, $C'(x_3',y_3')$.

Step2: Use mid - point formula

The line of reflection is the perpendicular bisector of the line segments joining corresponding points. For a point $P(x,y)$ and its reflection $P'(x',y')$ across a line, the mid - point of the line segment $PP'$ lies on the line of reflection. Let's take one pair of corresponding points, say $A$ and $A'$. The mid - point $M$ of the line segment $AA'$ has coordinates $M(\frac{x_1 + x_1'}{2},\frac{y_1 + y_1'}{2})$.

Step3: Determine slope of line joining corresponding points

Find the slope $m_{AA'}=\frac{y_1'-y_1}{x_1'-x_1}$ of the line segment joining $A$ and $A'$. The slope of the line of reflection is the negative reciprocal of $m_{AA'}$ (since the line of reflection is perpendicular to the line joining corresponding points).

Step4: Find equation of line of reflection

Using the mid - point and the slope of the line of reflection, we can find its equation. In the case of a grid - based problem like this, we can also observe visually. If we look at the grid, we can see that the line of reflection is a horizontal line. By counting the grid units between corresponding points, we find that the line of reflection is $y = 2$.

Answer:

The line of reflection is $y = 2$.