**Title: Understanding Translations in Coordinate Geometry****Introduction:**In coordinate geometry, a translation moves every point of a shape or object a constant distance in a specified direction. This is a common geometric transformation that preserves the shape and size of objects.**Example Problem:**Consider the following problem:- Point $ P'(-6, -4) $ is the image of point $ P(-2, 3) $ after a translation. What is the image of point $ A(5, -1) $ after you apply the same translation?**Steps to Solve the Problem:**1. **Determine the Translation Vector:** First, identify the change in coordinates from point $ P $ to point $ P' $. - Change in the x-coordinate: $ -6 - (-2) = -6 + 2 = -4 $ - Change in the y-coordinate: $ -4 - 3 = -7 $ Therefore, the translation vector is $( -4, -7 )$.2. **Apply the Translation Vector to Point A:** Use the translation vector to find the new coordinates for point $ A $. - New x-coordinate: $ 5 - 4 = 1 $ - New y-coordinate: $ -1 - 7 = -8 $ Thus, the image of point $ A(5, -1) $ after the translation is $ A'(1, -8) $.**Conclusion:**By determining the translation vector and applying it to a given point, we can find the resulting coordinates of the translated point. In this example, the image of point $ A(5, -1) $ after applying the same translation is $ A'(1, -8) $.**Interactive Tool:**Feel free to use the interactive tools provided below to experiment with translations by inputting different coordinates and observing the resulting transformations.**Formatting Options:**You can also format your results using bold, italics, or underline to emphasize important points.**Diagram:**Although this specific problem doesn't include a diagram, visualizing points and transformations on a coordinate plane can be helpful. Try plotting the points $ P(-2, 3) $, $ P'(-6, -4) $, $ A(5, -1) $, and $ A'(1, -8) \ **Reflection Over the X-Axis****Problem Statement:**Find the coordinates of the reflection without using the coordinate plane. Reflect point \( P(4,6) $ over the x-axis.**Solution:**To reflect a point over the x-axis, you keep the x-coordinate the same while changing the sign of the y-coordinate. For point $ P(4,6) $:1. Original coordinates of point $ P $: $ (4,6) $2. The reflection over the x-axis would yield the coordinates $ (4,-6) $.So, the coordinates of the reflection of point $ P(4,6) $ over the x-axis are $ (4,-6) $.

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Point P'(-6,-4) is the image of point P(-2,3) after a translation. What is the image of point A(5,-1) after you apply the same translation?

Elementary Geometry For College Students, 7e

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ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

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Concept explainers

Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.

Question

**Title: Understanding Translations in Coordinate Geometry**

**Introduction:**
In coordinate geometry, a translation moves every point of a shape or object a constant distance in a specified direction. This is a common geometric transformation that preserves the shape and size of objects.

**Example Problem:**
Consider the following problem:

- Point $ P'(-6, -4) $ is the image of point $ P(-2, 3) $ after a translation. What is the image of point $ A(5, -1) $ after you apply the same translation?

**Steps to Solve the Problem:**

1. **Determine the Translation Vector:**
First, identify the change in coordinates from point $ P $ to point $ P' $.

- Change in the x-coordinate: $ -6 - (-2) = -6 + 2 = -4 $
- Change in the y-coordinate: $ -4 - 3 = -7 $

Therefore, the translation vector is $( -4, -7 )$.

2. **Apply the Translation Vector to Point A:**
Use the translation vector to find the new coordinates for point $ A $.

- New x-coordinate: $ 5 - 4 = 1 $
- New y-coordinate: $ -1 - 7 = -8 $

Thus, the image of point $ A(5, -1) $ after the translation is $ A'(1, -8) $.

**Conclusion:**
By determining the translation vector and applying it to a given point, we can find the resulting coordinates of the translated point. In this example, the image of point $ A(5, -1) $ after applying the same translation is $ A'(1, -8) $.

**Interactive Tool:**
Feel free to use the interactive tools provided below to experiment with translations by inputting different coordinates and observing the resulting transformations.

**Formatting Options:**
You can also format your results using bold, italics, or underline to emphasize important points.

**Diagram:**
Although this specific problem doesn't include a diagram, visualizing points and transformations on a coordinate plane can be helpful. Try plotting the points $ P(-2, 3) $, $ P'(-6, -4) $, $ A(5, -1) $, and \( A'(1, -8) \

**Reflection Over the X-Axis**

**Problem Statement:**
Find the coordinates of the reflection without using the coordinate plane.
Reflect point $ P(4,6) $ over the x-axis.

**Solution:**

To reflect a point over the x-axis, you keep the x-coordinate the same while changing the sign of the y-coordinate. For point $ P(4,6) $:

1. Original coordinates of point $ P $: $ (4,6) $
2. The reflection over the x-axis would yield the coordinates $ (4,-6) $.

So, the coordinates of the reflection of point $ P(4,6) $ over the x-axis are $ (4,-6) $.

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