Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Understanding Translation
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Today, we're going to talk about translation, which is just a fancy way of saying slide. When we translate a shape, every point of that shape moves the exact same distance in the same direction. Can anyone explain what happens to the shape during a translation?
The shape stays the same, right? Like the size and angles don't change?
Exactly! The size, shape, and orientation remain unchanged. We only change the position. Now, let's look at how we represent this movement mathematically using vectors.
What do you mean by a vector?
Great question! A translation vector tells us how far, and in what direction, to move the shape. For instance, if we have a vector like (3, -1), it means we move the shape 3 units to the right and 1 unit down. Let's do an example together.
Working with Translation Vectors
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Let's practice translating triangle PQR with coordinates P(-2, 1), Q(0, 3), R(1, 0) using the vector (-1, 4). Who can tell me how we would find the new coordinates for point P?
We would subtract 1 from -2 and add 4 to 1.
Correct! So what does that give us?
That would be P'(-3, 5).
Great job! Now, who wants to calculate the coordinates for Q and R using the same vector?
For Q, it would be Q'(0-1, 3+4) = Q'(-1, 7) and for R it would be R'(1-1, 0+4) = R'(0, 4).
Excellent! Letโs summarize: when we translate, remember we apply the vector to each point individually. The new coordinates of P', Q', and R' are (-3, 5), (-1, 7), and (0, 4) respectively.
Practice Problems Review
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Now that we've completed translating triangle PQR, letโs tackle a new practice problem. Who's ready to translate rectangle WXYZ with vertices W(0, 4), X(3, 4), Y(3, 1), Z(0, 1) by the vector (-2, -3)?
I think we take 2 away from the x-coordinates and 3 away from the y-coordinates.
Exactly right! Can you show us what the new coordinates will be?
W'(-2, 1), X'(1, 1), Y'(1, -2), Z'(-2, -2).
Perfect! What do you notice about the size and shape of the rectangle after the translation?
Itโs the same shape but just moved down and left!
Well done, everyone! Remember, translations do not change the characteristics of the figure, just its position in space.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
This section presents practice problems that allow students to apply their knowledge of translations. Students will practice translating triangles and rectangles using given vectors while reinforcing their understanding of geometric transformations.
Detailed
Practice Problems 1.1
The purpose of Practice Problems 1.1 is to engage students in applying the concepts of translations in geometry. A translation is a type of geometric transformation that involves sliding a shape in the coordinate plane without rotating or flipping it. In this section, students will work with specific geometric figures: triangles and rectangles. They will receive specific translation vectors and will need to determine the new coordinates of the shapes after the transformation.
This practice reinforces the foundational concept that translations preserve the size and shape of the original figure, only altering its position. This knowledge is critical for mastering more complex transformations as students advance in geometric understanding.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Problem 1: Translating a Triangle
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
- Translate triangle PQR with vertices P(-2, 1), Q(0, 3), and R(1, 0) by the vector (-1, 4). Write the coordinates of the image P'Q'R'.
Detailed Explanation
In this problem, you have to translate a triangle named PQR. The translation vector given is (-1, 4). This means that for every point of the triangle, you will move it left by 1 unit (since it's negative) and up by 4 units. To find the new coordinates of each point after translation, follow these steps:
- For Point P(-2, 1):
- X-coordinate: -2 - 1 = -3
- Y-coordinate: 1 + 4 = 5
- Therefore, Point P' = (-3, 5)
- For Point Q(0, 3):
- X-coordinate: 0 - 1 = -1
- Y-coordinate: 3 + 4 = 7
- Therefore, Point Q' = (-1, 7)
- For Point R(1, 0):
- X-coordinate: 1 - 1 = 0
- Y-coordinate: 0 + 4 = 4
- Therefore, Point R' = (0, 4)
- The image of triangle PQR after translation will have vertices P'(-3, 5), Q'(-1, 7), and R'(0, 4).
Examples & Analogies
Imagine pushing your toy box across the floor. If it starts at a certain spot and you want to move it left by 1 foot and up by 4 feet, after you push it, it will end up in a new position, similar to how points on the triangle move to new coordinates.
Problem 2: Translating a Rectangle
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
- Rectangle WXYZ has vertices W(0, 4), X(3, 4), Y(3, 1), Z(0, 1). Translate it by (-2, -3). What are the coordinates of the image W'X'Y'Z'?
Detailed Explanation
Here, you need to translate rectangle WXYZ. The translation vector is (-2, -3), which means moving left by 2 units and down by 3 units for each vertex. To find the new coordinates for each point, do the following:
- For Point W(0, 4):
- X-coordinate: 0 - 2 = -2
- Y-coordinate: 4 - 3 = 1
- Thus, W' = (-2, 1)
- For Point X(3, 4):
- X-coordinate: 3 - 2 = 1
- Y-coordinate: 4 - 3 = 1
- Thus, X' = (1, 1)
- For Point Y(3, 1):
- X-coordinate: 3 - 2 = 1
- Y-coordinate: 1 - 3 = -2
- Thus, Y' = (1, -2)
- For Point Z(0, 1):
- X-coordinate: 0 - 2 = -2
- Y-coordinate: 1 - 3 = -2
- Thus, Z' = (-2, -2)
- After translating, the new vertices of rectangle WXYZ will be W'(-2, 1), X'(1, 1), Y'(1, -2), and Z'(-2, -2).
Examples & Analogies
Think of drawing a rectangle on a piece of paper. If you take that paper and slide it down and to the left without rotating it, each corner of your rectangle will end up at new locations on the paper.
Problem 3: Describing a Translation with a Vector
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
- A point M(5, -3) is translated to M'(-2, -7). Describe this translation using a column vector.
Detailed Explanation
In this problem, you need to figure out how the point M(5, -3) has moved to M'(-2, -7). To find the translation vector:
- Start by calculating the change in the x-coordinate:
- From 5 to -2: Change in x = -2 - 5 = -7. This means the point moved 7 units left.
- Now, calculate the change in the y-coordinate:
- From -3 to -7: Change in y = -7 - (-3) = -7 + 3 = -4. This means the point moved 4 units down.
- Thus, the translation vector is (-7, -4), written as a column vector:
[ -7
-4 ]
Examples & Analogies
Imagine a ship on a map. If it moves 7 miles to the left (west) and 4 miles down (south), you can think of each movement as a step in a direction, giving you the translation vector that describes its journey.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Translation: A movement of a shape on a coordinate plane where its size and shape do not change.
-
Vector: A mathematical representation of direction and magnitude for translating shapes.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Example of translating triangle PQR by a vector (-1, 4) yields new coordinates P'(-3, 5), Q'(-1, 7), R'(0, 4).
-
Example of translating rectangle WXYZ by vector (-2, -3) gives new vertices W'(-2, 1), X'(1, 1), Y'(1, -2), Z'(-2, -2).
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
๐ต Rhymes Time
-
When shapes slide with grace, they keep their place, size and all, just a move, that's the translation call.
๐ Fascinating Stories
-
Imagine a box on a slide, with no tilt, no frets, just gliding down smoothly to spots where it sets. That's a translation, just a smooth glide.
๐ง Other Memory Gems
-
SLIDE: Shift, Leave shape intact, Invert nothing, Direction specified, Every point moves the same.
๐ฏ Super Acronyms
TMR
- Translation Means Relocation.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Translation
Definition:
A transformation that slides a shape in a specific direction without changing its size or orientation.
-
Term: Coordinates
Definition:
Specific points defined by an ordered pair (x, y) representing positions on the coordinate plane.
-
Term: Vector
Definition:
A quantity that has both direction and magnitude, often represented as an ordered pair indicating movement.