Translations in Geometry
In geometry, a translation is a type of transformation that moves every point of a shape or object the same distance in the same direction. This lesson will help you understand how translations work, with clear examples and exercises to practice.
Outline
- Definition of Translation
- Examples of Translations
- Exercises with Solutions
- Homework
- Revision Section
- Embedded Video
1. Definition of Translation
A translation slides a shape or object from one location to another without rotating, resizing, or flipping it. Each point of the shape moves the same distance in the same direction.
2. Examples of Translations
Example 1: Translating a Point
Translate the point \( (2, 3) \) 4 units to the right and 2 units down. What are the new coordinates?
New coordinates: \( (2 + 4, 3 - 2) = (6, 1) \)
Example 2: Translating a Triangle
Translate triangle \( \triangle ABC \) with vertices \( A(1, 2) \), \( B(3, 2) \), and \( C(2, 4) \) 5 units left and 3 units up. What are the new coordinates?
New coordinates:
- A'(-4, 5)
- B'(-2, 5)
- C'(-3, 7)
Example 3: Translating a Rectangle
Translate rectangle \( ABCD \) with vertices \( A(0, 0) \), \( B(0, 3) \), \( C(4, 3) \), and \( D(4, 0) \) 3 units to the right and 1 unit down. What are the new coordinates?
New coordinates:
- A'(3, -1)
- B'(3, 2)
- C'(7, 2)
- D'(7, -1)
Example 4: Translating a Polygon
Translate the pentagon with vertices \( (2, 3) \), \( (4, 3) \), \( (5, 5) \), \( (3, 6) \), and \( (1, 5) \) 2 units to the left and 4 units up. What are the new coordinates?
New coordinates:
- (0, 7)
- (2, 7)
- (3, 9)
- (1, 10)
- (-1, 9)
Example 5: Translating Coordinates in a Grid
Translate the coordinates \( (5, 7) \), \( (8, 7) \), and \( (8, 9) \) 6 units to the right and 2 units down. What are the new coordinates?
New coordinates:
- (11, 5)
- (14, 5)
- (14, 7)
Example 6: Translating an Object on a Coordinate Plane
A shape with vertices \( (3, 2) \), \( (5, 2) \), \( (5, 4) \), and \( (3, 4) \) is translated 1 unit left and 6 units up. What are the new coordinates?
New coordinates:
- (2, 8)
- (4, 8)
- (4, 10)
- (2, 10)
3. Exercises with Solutions
Exercise 1
Translate the point \( (-1, -2) \) 3 units to the right and 4 units up. What are the new coordinates?
New coordinates: \( (-1 + 3, -2 + 4) = (2, 2) \)
Exercise 2
Translate the square with vertices \( (0, 0) \), \( (0, 2) \), \( (2, 2) \), and \( (2, 0) \) 5 units left and 3 units up. What are the new coordinates?
New coordinates:
- (-5, 3)
- (-5, 5)
- (-3, 5)
- (-3, 3)
Exercise 3
Translate the triangle \( \triangle XYZ \) with vertices \( (1, 1) \), \( (4, 1) \), and \( (2, 4) \) 2 units to the right and 3 units down. What are the new coordinates?
New coordinates:
- (3, -2)
- (6, -2)
- (4, 1)
Exercise 4
Translate the hexagon with vertices \( (2, 1) \), \( (4, 1) \), \( (5, 2) \), \( (4, 3) \), \( (2, 3) \), and \( (1, 2) \) 3 units to the left and 2 units down. What are the new coordinates?
New coordinates:
- (-1, -1)
- (1, -1)
- (2, 0)
- (1, 1)
- (-1, 1)
- (-2, 0)
4. Homework
Complete the following exercises:
- Translate the point \( (4, -3) \) 5 units to the left and 2 units up.
- Translate the rectangle with vertices \( (1, 2) \), \( (1, 5) \), \( (4, 5) \), and \( (4, 2) \) 3 units to the right and 1 unit down.
- Translate the pentagon with vertices \( (0, 0) \), \( (1, 1) \), \( (2, 0) \), \( (2, 2) \), and \( (1, 3) \) 2 units to the right and 4 units down.
5. Revision
Review the key concepts of translations:
- Translations shift every point of a shape in the same direction by the same distance.
- The shape and size of the object remain unchanged after translation.
- Translations can be applied to any geometric figure or coordinate point.
Need more help? Contact us for online tutoring or text us to book a tutor. Don’t forget to follow us on Facebook and YouTube.
0 Comments