https://file.aiquickdraw.com/mathbot/file/1726639308231duv6xuj5.jpg

To solve this problem, let's break it down step by step.

### Original Coordinates of the Rectangle
- \( A(2, 3) \)
- \( B(4, 3) \)
- \( C(4, -1) \)
- \( D(2, -1) \)

### Rotation of 180° about the Origin
When you rotate a point \((x, y)\) by 180° about the origin, the coordinates of the point become \((-x, -y)\).

Now, apply this transformation to the given points:

- \( A(2, 3) \) becomes \( A'(-2, -3) \)
- \( B(4, 3) \) becomes \( B'(-4, -3) \)
- \( C(4, -1) \) becomes \( C'(-4, 1) \)
- \( D(2, -1) \) becomes \( D'(-2, 1) \)

### Evaluating the Questions

1. **Length of Corresponding Segments**: 
   The lengths of the sides of a rectangle do not change when rotated. Thus, the length of the corresponding segments remains the **same**.

2. **Measurements of Corresponding Angles**: 
   Since rotation preserves the angles, the corresponding angles will remain the **same**.

3. **Signs of the X-coordinates**: 
   The signs of the X-coordinates will change because each coordinate is multiplied by \(-1\). Thus, the signs of the X-coordinates will be **different**.

4. **Signs of the Y-coordinates**: 
   Similarly, the Y-coordinates are also multiplied by \(-1\), so the signs of the Y-coordinates will be **different**.

### Summary
- **Length of corresponding segments**: Same
- **Measurements of corresponding angles**: Same
- **Signs of the X-coordinates**: Different
- **Signs of the Y-coordinates**: Different

Would you like more details or have any questions? Here are 5 related questions to expand your understanding:

1. How do rotations affect distances between points?
2. What are the properties of a rectangle after rotation by 90°?
3. How does a reflection differ from a rotation in terms of coordinates?
4. What would happen to the coordinates if the figure was rotated by 90° instead of 180°?
5. How would the coordinates change if the rectangle was rotated around a point other than the origin?

**Tip:** Always remember that a 180° rotation around the origin results in the negation of both x- and y-coordinates.

A rectangle has coordinates A(2, 3), B(4, 3), C(4, -1), and D(2, -1). The figure is rotated 180° with the origin as the center of rotation to produce a new figure. For each statement, determine if the new rotated figure has the same or different qualities as the original figure: length of corresponding segments, measurements of corresponding angles, signs of the X-coordinates, signs of the Y-coordinates.

Solve a geometry problem where a rectangle with coordinates A(2, 3), B(4, 3), C(4, -1), and D(2, -1) is rotated by 180° about the origin. Explore how the lengths, angles, and signs of the coordinates change with this transformation. Ideal 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