Question

The coordinates of a triangle are P(1, 4), Q(3, 6), and R(5, 2). The triangle is reflected over a line and its image coordinates are P'(–1, 4), Q'(–3, 6), and R'(–5, 2).



Find the equation of the reflection line.


x = –1
y = –1
y = 0
x = 0

Answer 1

To find the equation of the reflection line over which the triangle is reflected, we will use the properties of reflection in geometry.

1. **Identify the Midpoints**:
The coordinates of the original points are \(P(1, 4)\), \(Q(3, 6)\), and \(R(5, 2)\), and the coordinates of the reflected points are \(P'(-1, 4)\), \(Q'(-3, 6)\), and \(R'(-5, 2)\).

- For point \(P\) and \(P'\):
- Midpoint \(M_P = \left(\frac{1 + (-1)}{2}, \frac{4 + 4}{2}\right) = (0, 4)\)

- For point \(Q\) and \(Q'\):
- Midpoint \(M_Q = \left(\frac{3 + (-3)}{2}, \frac{6 + 6}{2}\right) = (0, 6)\)

- For point \(R\) and \(R'\):
- Midpoint \(M_R = \left(\frac{5 + (-5)}{2}, \frac{2 + 2}{2}\right) = (0, 2)\)

2. **Analyzing the Midpoints**:
All the midpoints \(M_P\), \(M_Q\), and \(M_R\) have the same x-coordinate of \(0\) (which means they are on the line \(x = 0\)). This suggests that the line of reflection is vertical, possibly along the y-axis.

3. **Conclusion**:
Since the coordinates of the midpoints of the segments connecting each original point and its image all lie on the line \(x = 0\), the line of reflection is indeed the **y-axis**.

Therefore, the equation of the reflection line is:

**x = 0**.