To find the coordinates of the point \( A' \) that is the reflection of point \( A(1, 2) \) across line \( AB \) defined by points \( A(1, 2) \) and \( B(-2, -1) \), we first need to determine the equation of line \( AB \), then find the perpendicular line from point \( A \) to line \( AB \), and finally calculate the reflection point \( A' \).
Step 1: Find the equation of line \( AB \)
The slope \( m \) of line \( AB \) can be calculated using the coordinates of points \( A \) and \( B \):
\[ m = \frac{y_B - y_A}{x_B - x_A} = \frac{-1 - 2}{-2 - 1} = \frac{-3}{-3} = 1 \]
Now, using the point-slope form of the line equation \( y - y_1 = m(x - x_1) \) with point \( A(1, 2) \):
\[ y - 2 = 1(x - 1) \] \[ y - 2 = x - 1 \] \[ y = x + 1 \]
Step 2: Find the perpendicular line from point \( A(1,2) \)
The slope of the perpendicular line (normal line to line \( AB \)) will be the negative reciprocal of the slope of line \( AB \):
\[ m_{\text{perpendicular}} = -1 \]
Using point \( A(1, 2) \) in the point-slope form:
\[ y - 2 = -1(x - 1) \] \[ y - 2 = -x + 1 \] \[ y = -x + 3 \]
Step 3: Find the intersection of the two lines
Now, we need to find the point of intersection between \( y = x + 1 \) and \( y = -x + 3 \):
Setting the equations equal to each other:
\[ x + 1 = -x + 3 \] \[ 2x = 2 \implies x = 1 \]
Now, substitute \( x = 1 \) back into either equation to find \( y \):
\[ y = 1 + 1 = 2 \]
Thus, the intersection point (which is also point \( A \)) is \( I(1, 2) \).
Step 4: Finding point \( A' \)
The coordinates of point \( A' \) will be such that they are equidistant from point \( I \) as point \( A \) but on the opposite side of line \( AB \).
Since \( A \) and \( I \) are the same point \( (1, 2) \), we simply need to "move" to the same distance below line \( AB \).
The reflection point \( A' \) can be calculated directly by using the direction of the perpendicular and moving the same distance again. From point \( I(1, 2) \) on line AB to point \( A' \):
To find point \( A' \):
A reflection across line \( AB \): You can use the midpoint and then calculate \( d \):
- The distance vector from \( I \) to \( A \) is \( (0, 0) \) since they are the same, so we don't need to calculate anything here because they are equal.
Coordinates of reflection point \( A' \)
The symmetric point \( A' \) about line \( AB \) (as it is on the line itself) will maintain the vertical distance between point \( I \) and \( line AB \):
The reflection point \( A' \) is at:
\[ A' = (1 - 3, 2 - 3) = (-2, -1) \]
Thus, the coordinates of \( A' \) are \(\textbf{A'}(3, 4)\).