Reflections Over the Line x=2: A Geometry Lesson

In geometry, reflections are transformations that flip a figure over a line, known as the line of reflection. This creates a mirror image of the original figure. Today, we'll delve into reflections over the line x=2. Let's explore how to reflect figures over this specific line and understand the key concepts involved.

Understanding Reflections

Imagine holding a mirror up to a shape. The image you see in the mirror is a reflection of the original shape. In geometry, we use the same concept. A reflection over a line creates a mirror image of the original figure, where every point in the original figure is the same distance from the line of reflection as its corresponding point in the reflected figure.

Reflecting a Triangle Over x=2

Let's consider a triangle with vertices A(1,1), B(3,3), and C(2,5). We want to reflect this triangle over the line x=2. Here's how we do it:

1. **Identify the line of reflection:** In this case, it's the vertical line x=2.
2. **Find the distance from each vertex to the line x=2:** For example, point A is 1 unit to the left of the line x=2. Therefore, its reflected point will be 1 unit to the right of the line x=2.
3. **Reflect each vertex:** For point A(1,1), the reflected point A' will be (3,1). Similarly, B(3,3) will reflect to B'(1,3), and C(2,5) will reflect to C'(2,5).
4. **Connect the reflected points:** Connect A', B', and C' to form the reflected triangle.

Notice that the reflected triangle is congruent to the original triangle, meaning they have the same size and shape. However, they are mirror images of each other.

Reflecting a Line Over x=2

Let's take a line segment with endpoints D(1,4) and E(4,1). To reflect this line over x=2, we follow the same steps as above:

1. **Identify the line of reflection:** x=2.
2. **Find the distance from each endpoint to x=2:** D is 1 unit to the left of x=2, and E is 2 units to the right.
3. **Reflect each endpoint:** D' will be (3,4) and E' will be (0,1).
4. **Connect the reflected endpoints:** Connect D' and E' to form the reflected line.

The reflected line segment is also congruent to the original line segment but is flipped over the line of reflection.

Key Concepts

• **Line of reflection:** The line over which the figure is flipped.
• **Distance from a point to a line:** The perpendicular distance from the point to the line.
• **Congruence:** Two figures are congruent if they have the same size and shape.

Practice Questions

1. Reflect the point (5,2) over the line x=2. What are the coordinates of the reflected point?

2. Reflect the rectangle with vertices (1,1), (4,1), (4,3), and (1,3) over the line x=2. Sketch the reflected rectangle.

By understanding reflections over the line x=2, you gain valuable insights into geometric transformations. These concepts are fundamental in various fields, including art, architecture, and engineering.