Translations are transformations that move points or objects in a plane without changing their shape or orientation. This section focuses on understanding translations through vector notation, crucial for IGCSE students mastering geometry.
Introduction to Translations
In geometry, a translation moves a figure or point from one location to another on the plane. This movement is described using vectors, providing a clear mathematical representation of the translation process.
![Translation](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2Ff35f4542-d852-4328-91fb-bb41c6d14e2a-file.png&w=1080&q=75)
Image courtesy of Online Math Learning
Understanding Vector Notation
Vector Basics
- A vector represents both direction and magnitude.
- In 2D, vectors are written as , where and indicate horizontal and vertical movements.
![Vectors](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2Fab845df8-641f-491f-8621-f7fc41906359-file.png&w=1080&q=75)
Image courtesy of Cuemath
Vectors and Translations
- A translation vector shows the shift from point to point .
- and detail the movement along the x-axis and y-axis.
Applying Translations
Translation Formula
For any point translated by , the new position is calculated as:
Worked Examples
Translating a Point
Given: Point , Vector
Find: New position
Solution:
Hence, moves to.
![Vector and Translation](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F5c77923f-9264-4a3e-aa94-14306ddd7cfd-file.png&w=1080&q=75)
Translating a Shape
Given: Triangle vertices , , , Vector
Find: New vertices positions
Solution:
- :
- :
- :
New vertices: , , .
![Translating Triangle](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F715a16b9-de55-4996-a3f9-ddc095dc4208-file.png&w=1080&q=75)
Practice Problems
Problem 1
Given: ,
Find:
Solution:
![Translating a Point](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F10fb2397-863f-4e4d-8a36-b392b6579ca8-file.png&w=1080&q=75)
Problem 2
Given: Square vertices, , , ,
Find: New vertices positions
Solution:
- :
- :
- :
- :
New square vertices: , , , .
![Translating Square](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2Fce7b6a03-f220-4971-826c-7f12c1afca29-file.png&w=1080&q=75)
Success Tips
- Draw diagrams to visualize translations.
- Solve various problems for practice.
- Ensure understanding of vector operations (addition, scalar multiplication).