Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Title: Understanding Reflections in Coordinate Geometry**
**Objective:** Determine the equation of the line of reflection for a given geometric transformation on a coordinate plane.
**Problem Statement:**
- "Write the equation of the line of reflection."
**Diagrams and Analysis:**
In the image provided, we have two quadrilaterals plotted on a coordinate grid, which are reflections of each other. The vertices of these quadrilaterals are labeled as \( W, X, Y, Z \) and \( W', X', Y', Z' \).
**Quadrilateral Details:**
1. **Original Quadrilateral (WXYZ) Coordinates:**
- \( W(-6, -4) \)
- \( X(-4, 0) \)
- \( Y(-2, 2) \)
- \( Z(-5, 1) \)
2. **Reflected Quadrilateral (W'X'Y'Z') Coordinates:**
- \( W'(-4, -6) \)
- \( X'(0, -4) \)
- \( Y'(2, -2) \)
- \( Z'(1, -5) \)
**Line of Reflection:**
To determine the line of reflection, observe the midpoints of corresponding points from the original and the reflected quadrilateral. Each midpoint should lie on this line of reflection.
- Midpoint between \( W(-6, -4) \) and \( W'(-4, -6) \):
\[
\left( \frac{-6 + (-4)}{2}, \frac{-4 + (-6)}{2} \right) = (-5, -5)
\]
- Midpoint between \( X(-4, 0) \) and \( X'(0, -4) \):
\[
\left( \frac{-4 + 0}{2}, \frac{0 + (-4)}{2} \right) = (-2, -2)
\]
Following similar calculations for other points suggest that the line of reflection is a diagonal line that can be written as the line \( y = x \).
**Conclusion:**
The equation of the line of reflection for the given transformation is:
\[ y = -x \]
**Note:** Please ensure to input equations without spaces as per the instruction: "Do not type any spaces between the characters in your equation](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd2510d80-3418-456e-8281-5d18c2652c22%2F37b4be7f-b9e8-46ae-9938-83a4a997cb00%2Fiztbhtv_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Reflections in Coordinate Geometry**
**Objective:** Determine the equation of the line of reflection for a given geometric transformation on a coordinate plane.
**Problem Statement:**
- "Write the equation of the line of reflection."
**Diagrams and Analysis:**
In the image provided, we have two quadrilaterals plotted on a coordinate grid, which are reflections of each other. The vertices of these quadrilaterals are labeled as \( W, X, Y, Z \) and \( W', X', Y', Z' \).
**Quadrilateral Details:**
1. **Original Quadrilateral (WXYZ) Coordinates:**
- \( W(-6, -4) \)
- \( X(-4, 0) \)
- \( Y(-2, 2) \)
- \( Z(-5, 1) \)
2. **Reflected Quadrilateral (W'X'Y'Z') Coordinates:**
- \( W'(-4, -6) \)
- \( X'(0, -4) \)
- \( Y'(2, -2) \)
- \( Z'(1, -5) \)
**Line of Reflection:**
To determine the line of reflection, observe the midpoints of corresponding points from the original and the reflected quadrilateral. Each midpoint should lie on this line of reflection.
- Midpoint between \( W(-6, -4) \) and \( W'(-4, -6) \):
\[
\left( \frac{-6 + (-4)}{2}, \frac{-4 + (-6)}{2} \right) = (-5, -5)
\]
- Midpoint between \( X(-4, 0) \) and \( X'(0, -4) \):
\[
\left( \frac{-4 + 0}{2}, \frac{0 + (-4)}{2} \right) = (-2, -2)
\]
Following similar calculations for other points suggest that the line of reflection is a diagonal line that can be written as the line \( y = x \).
**Conclusion:**
The equation of the line of reflection for the given transformation is:
\[ y = -x \]
**Note:** Please ensure to input equations without spaces as per the instruction: "Do not type any spaces between the characters in your equation
Transcribed Image Text:**Title: Understanding Reflections in Geometry**
**Objective: Learn how to find the equation of a line of reflection for a given geometric figure.**
**Text:**
**Write the equation of the line of reflection.**
**Diagram Description:**
The provided diagram consists of a grid with two parallelograms. The grid has an x-axis and a y-axis. Each square on the grid represents one unit.
- **Coordinates:**
- The upper parallelogram has vertices labeled G, H, J, and K.
- G is located at (0, 3), H at (2, 4), J at (6, 4), and K at (4, 3).
- The lower parallelogram has vertices labeled R, Q, S, and P.
- R is located at (2, 1), Q at (0, 0), S at (4, 0), and P at (6, 1).
**Line of Reflection:**
The line of reflection is horizontal, parallel to the x-axis. It is equidistant from the corresponding points on the two parallelograms. In this diagram, the line of reflection is the line y = 2.
Reflective symmetry involves flipping a shape over a line so that the shape appears unchanged, just mirrored. Understanding how to find the line of reflection is crucial in various fields, including geometry and design. Practice by identifying lines of reflection with different geometric figures to enhance your skills.
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