https://file.aiquickdraw.com/mathbot/file/1728511602776kwpytpk4.jpg

Let's break down the problem step by step.

### Problem Summary:
You are working with a regular pentagon \( ABCDE \).

1. **Part (a):** Draw the translation \( T_{EB} \), where each vertex of the pentagon is translated. Label the images of each vertex after the translation.
   - This requires you to apply the translation vector from \( E \) to \( B \) on all vertices of the pentagon.

2. **Part (b):** Investigate whether a translation can map more than one point of the pentagon onto the same point. If possible, explain why.

### Part (a): Understanding Translation \( T_{EB} \)
The vector \( T_{EB} \) describes a translation from point \( E \) to point \( B \). When applying this translation to each vertex:
- Translate \( A \), \( B \), \( C \), \( D \), and \( E \) by the same vector as \( E \to B \).
- For each vertex \( P \), the image will be \( P' = P + \overrightarrow{EB} \).

#### Translated Image of Each Vertex:
- \( A' = A + \overrightarrow{EB} \)
- \( B' = B + \overrightarrow{EB} \)
- \( C' = C + \overrightarrow{EB} \)
- \( D' = D + \overrightarrow{EB} \)
- \( E' = E + \overrightarrow{EB} = B \) (since this is the exact translation vector)

### Part (b): Investigating Multiple Points Mapping to the Same Point
In a translation, every point in the figure is moved by the same distance in the same direction. Therefore, each point \( P \) has a unique translation \( P' \), which means that **no two distinct points of the pentagon can map onto the same point** under a translation. This is because translations are rigid motions that preserve distance and do not "collapse" points together.

So, **it is not possible for a translation to map more than one point of the pentagon onto the same point**.

### Follow-Up Questions:
1. How is the translation vector \( \overrightarrow{EB} \) determined geometrically?
2. What would the result be if the pentagon was reflected instead of translated?
3. What happens if the translation vector \( T_{EB} \) is applied twice in succession? Would it still map the pentagon to a new location?
4. Can a rotation about the center of the pentagon map two vertices onto the same point?
5. What is the difference between a translation and other types of transformations like rotation or reflection?

#### Tip:
When applying transformations like translations, always remember that distances between points are preserved, so no point can overlap with another under such a transformation.

The diagram shows regular pentagon ABCDE. a. Draw the translation T_EB. Label the images of each vertex. b. Is it possible for a translation to map more than one point of the pentagon onto another point of the pentagon? Explain.

Explore how translations work in a regular pentagon with vector T_EB. Learn how to label the images of each vertex and investigate whether a translation can map multiple points to the same location. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation