**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) $.

Hey, since there are multiple questions posted, we will answer first question. If you want any…

Answered: Point P'(-6,-4) is the image of point…

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

7th Edition

ISBN:9781337614085

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

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

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT

See similar textbooks

Related questions

Q: (c)

Q: Find the distance between (10,-11) and (12,3

Q: uate ∫c 5x^4 ds, where C is the line segment from (1,3) to(2,6).

A: In this question, the concept of line integral is applied. Line Integral The function to be…

Q: Point a is at (-3,-5) and point b is at (1,-9) what is the midpoint of the line segment ab

Q: The distance between the points (-2,2,0) and (1, -1,0) is 3/2. True False

Q: Point P' (-6, - 4) is the image of point P (-2, 3) after a translation. What is the image of the…

A: Let the translation be Tx,y=x+a, y+b Now T-6,4=(-2,3)⇒-6+a, -4+b=(-2,3)(a,b)=(-2+6, 3+4)=(4,7)

Q: Can you find the distance between the line x-9=y-7=z-10 and the line x-11=y-11=3z-19?

A: detail solution written below

Q: Is the line through the points (3, 4) and (-1, 2) parallel to the line 2x + 3y = 0? Justify your…

A: Given two points: 3, 4 and -1, 2 ……1 Given line: 2x+3y=0 ……2

Q: Find the area and perimeter is x=4.2 pls

A: if x = 4.2 9x=9·4.2=37.8 2x-5=2· 4.2-5=3.4

Q: What is the distance between the points ( 7 , − 1 ) and ( − 5 , 4 )

Q: Find the distance of the line segment that goes from (0,2) to (1,7)

Q: What's the distance between the point (-9,2) and (-4,2)?

A: Given : point (-9,2) and (-4,2)

Q: Verify that (3, 1) is the midpoint of the line segment joining (-2, 6) and (8, -4).

A: Consider a line segment AB such that A=x1, y1 & B=x2, y2 Let P be the midpoint of the line…

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) $.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Cartesian Coordinates$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, geometry and related others by exploring similar questions and additional content below.

Recommended textbooks for you

Elementary Geometry For College Students, 7e

Geometry

ISBN:

9781337614085

Author:

Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:

Cengage,