Finding The Reflection That Maps A Line Segment From (-1, 4) And (4, 1) To (-4, 1) And (-1, -4)

by Scholario Team 96 views

Hey guys! Today, we're diving into a fun geometry problem that involves reflections. Specifically, we want to figure out which reflection will transform a line segment with endpoints at (-1, 4) and (4, 1) to a new line segment with endpoints at (-4, 1) and (-1, -4). Sounds like a puzzle, right? Let's break it down step by step.

Understanding Reflections

Before we jump into solving this problem, let's quickly recap what reflections are. In geometry, a reflection is a transformation that creates a mirror image of a shape or a point across a line, which we call the line of reflection. Think of it like folding a piece of paper along the line of reflection – the image on one side is the flipped version of the image on the other side.

There are a few common types of reflections we often encounter:

  • Reflection across the x-axis: When we reflect a point across the x-axis, the x-coordinate stays the same, but the y-coordinate changes its sign. So, a point (x, y) becomes (x, -y).
  • Reflection across the y-axis: Reflecting across the y-axis means the y-coordinate stays the same, while the x-coordinate changes its sign. A point (x, y) becomes (-x, y).
  • Reflection across the line y = x: This reflection swaps the x and y coordinates. A point (x, y) becomes (y, x).
  • Reflection across the line y = -x: This reflection swaps the x and y coordinates and changes the signs of both. A point (x, y) becomes (-y, -x).

Knowing these rules is super helpful for tackling reflection problems. Now, let's get back to our specific question.

Analyzing the Endpoint Transformations

Our line segment initially has endpoints at (-1, 4) and (4, 1). After the reflection, the endpoints are at (-4, 1) and (-1, -4). To figure out which reflection occurred, we need to carefully analyze how the coordinates changed.

Let's look at the first endpoint, (-1, 4), and see how it transformed into (-4, 1). Notice that the x-coordinate changed from -1 to -4, and the y-coordinate changed from 4 to 1. This doesn't directly match any of our standard reflection rules (x-axis, y-axis, y=x, or y=-x) at first glance. However, let's keep this in mind as we look at the second endpoint.

Now, let's examine the second endpoint, (4, 1), which transformed into (-1, -4). Here, the x-coordinate changed from 4 to -1, and the y-coordinate changed from 1 to -4. This transformation looks similar to the first one, but the changes are in the opposite direction. Again, this doesn't immediately fit our standard reflection rules.

What's happening here? It seems like a single reflection across one of the standard axes isn't enough to explain the transformation. We need to think outside the box a little.

Testing Reflection Options

Since we're given answer choices, let's test them out and see which one works. This is a great strategy for multiple-choice questions!

A. Reflection Across the y-axis

If we reflect the original points across the y-axis, the x-coordinates change signs, and the y-coordinates stay the same.

  • (-1, 4) would become (1, 4)
  • (4, 1) would become (-4, 1)

This doesn't match our target endpoints of (-4, 1) and (-1, -4), so option A is not the correct answer.

B. Reflection Across the x-axis

Reflecting across the x-axis changes the sign of the y-coordinates while keeping the x-coordinates the same.

  • (-1, 4) would become (-1, -4)
  • (4, 1) would become (4, -1)

Again, this doesn't match our target endpoints, so option B is also incorrect.

C. Reflection Across the Line y = x

Reflecting across the line y = x swaps the x and y coordinates.

  • (-1, 4) would become (4, -1)
  • (4, 1) would become (1, 4)

Still not a match! Let's keep going.

D. Reflection Across the Line y = -x

When reflecting across the line y = -x, we swap the x and y coordinates and change the signs of both.

  • (-1, 4) would become (-4, 1)
  • (4, 1) would become (-1, -4)

Bingo! This matches our target endpoints perfectly. So, the correct answer is a reflection across the line y = -x.

The Solution

After analyzing the transformations and testing the reflection options, we found that a reflection across the line y = -x will produce an image with endpoints at (-4, 1) and (-1, -4).

Key Takeaways

This problem highlights a few important concepts:

  • Understanding reflections: Knowing how reflections work across different lines (x-axis, y-axis, y = x, y = -x) is crucial.
  • Analyzing coordinate changes: Carefully observing how the coordinates change can give you clues about the transformation.
  • Testing options: When you're stuck, especially in multiple-choice questions, try testing the answer choices.

Geometry problems can seem intimidating at first, but by breaking them down into smaller steps and understanding the underlying concepts, you can conquer them! Keep practicing, and you'll become a reflection master in no time. Great job, guys! You nailed it!