Reflections in Geometry
In geometry, a reflection is a transformation that flips a shape or point over a line, creating a mirror image. This lesson covers the basics of reflecting points and shapes across a line, with detailed examples and exercises to practice.
Outline
- Definition of Reflection
- Examples of Reflections
- Exercises with Solutions
- Homework
- Revision Section
- Embedded Video
1. Definition of Reflection
A reflection in geometry is a transformation that creates a mirror image of a shape or point across a specific line, called the line of reflection. Each point of the shape or object is flipped over the line, maintaining equal distances from the line of reflection.
Example of a reflection across a vertical line.
2. Examples of Reflections
Example 1: Reflecting a Point
Reflect the point \( (3, 2) \) across the line \( x = 1 \). What are the coordinates of the reflected point?
To find the reflected point, use the formula for reflection over a vertical line: \[ (x', y) = (2a - x, y) \] where \( a \) is the x-coordinate of the line of reflection. For \( x = 1 \): \[ x' = 2 \cdot 1 - 3 = -1 \] Thus, the reflected point is \( (-1, 2) \).
Example 2: Reflecting Across a Horizontal Line
Reflect the point \( (4, -3) \) across the line \( y = 2 \). What are the coordinates of the reflected point?
To find the reflected point over a horizontal line: \[ (x, y') = (x, 2b - y) \] where \( b \) is the y-coordinate of the line of reflection. For \( y = 2 \): \[ y' = 2 \cdot 2 - (-3) = 7 \] Thus, the reflected point is \( (4, 7) \).
Example 3: Reflecting a Triangle
Reflect triangle \( \triangle ABC \) with vertices \( A(1, 1) \), \( B(3, 1) \), and \( C(2, 4) \) across the line \( y = 2 \). What are the new coordinates?
Reflecting each point:
- A'(1, 2 \cdot 2 - 1) = (1, 3)
- B'(3, 2 \cdot 2 - 1) = (3, 3)
- C'(2, 2 \cdot 2 - 4) = (2, 0)
Example 4: Reflecting a Rectangle
Reflect rectangle \( ABCD \) with vertices \( A(0, 0) \), \( B(0, 2) \), \( C(4, 2) \), and \( D(4, 0) \) across the line \( x = 2 \). What are the new coordinates?
Reflecting each point:
- A'(2 \cdot 2 - 0, 0) = (4, 0)
- B'(2 \cdot 2 - 0, 2) = (4, 2)
- C'(2 \cdot 2 - 4, 2) = (0, 2)
- D'(2 \cdot 2 - 4, 0) = (0, 0)
Example 5: Reflecting a Pentagon
Reflect pentagon with vertices \( (1, 1) \), \( (2, 3) \), \( (4, 3) \), \( (5, 2) \), and \( (3, 1) \) across the line \( y = 1 \). What are the new coordinates?
Reflecting each point:
- (1, 2 \cdot 1 - 1) = (1, 1)
- (2, 2 \cdot 1 - 3) = (2, -1)
- (4, 2 \cdot 1 - 3) = (4, -1)
- (5, 2 \cdot 1 - 2) = (5, 0)
- (3, 2 \cdot 1 - 1) = (3, 1)
Example 6: Reflecting Across the Origin
Reflect the point \( (-2, 3) \) across the origin. What are the coordinates of the reflected point?
To reflect across the origin, negate both coordinates: \[ (-x, -y) = (2, -3) \] Thus, the reflected point is \( (2, -3) \).
3. Exercises
Reflect the following shapes across the given lines and provide the coordinates of the reflected shapes:
- Point \( (5, -2) \) across the line \( x = 0 \).
- Triangle \( \triangle DEF \) with vertices \( D(-1, -1) \), \( E(-4, -1) \), and \( F(-1, -5) \) across the line \( y = -2 \).
- Square with vertices \( (0, 0) \), \( (0, 3) \), \( (3, 3) \), and \( (3, 0) \) across the line \( x = 1.5 \).
Solutions:
- Point \( (5, -2) \) reflected across \( x = 0 \) is \( (-5, -2) \).
- Triangle \( \triangle DEF \) reflected across \( y = -2 \) has vertices \( D(-1, -3) \), \( E(-4, -3) \), and \( F(-1, -1) \).
- Square reflected across \( x = 1.5 \) has vertices \( (3, 0) \), \( (3, 3) \), \( (0, 3) \), and \( (0, 0) \).
4. Homework
Complete the following exercises:
- Reflect the point \( (-3, 4) \) across the line \( y = 0 \).
- Reflect the rectangle with vertices \( (1, 1) \), \( (1, 5) \), \( (4, 5) \), and \( (4, 1) \) across the line \( x = 2.5 \).
- Reflect the hexagon with vertices \( (2, 2) \), \( (4, 2) \), \( (5, 3) \), \( (5, 5) \), \( (4, 6) \), and \( (2, 6) \) across the line \( y = 4 \).
5. Revision
Review the key concepts of reflections:
- A reflection flips a shape or point over a line, creating a mirror image.
- The distance between the original shape and the line of reflection is equal to the distance between the reflected shape and the line.
- Reflections maintain the size and shape of the original figure but change its orientation.
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