Answered

Discover a wealth of knowledge and get your questions answered on IDNLearn.com. Get prompt and accurate answers to your questions from our community of experts who are always ready to help.

Rectangle ABCD has vertices [tex]\( A(-6,-2), B(-3,-2), C(-3,-6), \)[/tex] and [tex]\( D(-6,-6) \)[/tex]. The rectangle is translated so that the coordinates of the image are [tex]\( A'(-10,1), B'(-7,1), C'(-7,-3), \)[/tex] and [tex]\( D'(-10,-3) \)[/tex].

Which rule was used to translate the image?

A. [tex]\( T_{-4,3}(x, y) \)[/tex]

B. [tex]\( T_{-4,1}(x, y) \)[/tex]

C. [tex]\( T_{4,-1}(x, y) \)[/tex]

D. [tex]\( T_{4,-3}(x, y) \)[/tex]

Sagot :

GinnyAnsweridn
Let's solve the problem step-by-step to determine which translation rule was applied.

### Step 1: List the Original and Translated Coordinates
The vertices of the rectangle before the translation are:
- [tex]\(A = (-6, -2)\)[/tex]
- [tex]\(B = (-3, -2)\)[/tex]
- [tex]\(C = (-3, -6)\)[/tex]
- [tex]\(D = (-6, -6)\)[/tex]

The vertices of the rectangle after the translation are:
- [tex]\(A' = (-10, 1)\)[/tex]
- [tex]\(B' = (-7, 1)\)[/tex]
- [tex]\(C' = (-7, -3)\)[/tex]
- [tex]\(D' = (-10, -3)\)[/tex]

### Step 2: Calculate the Translation Vector
To find the translation vector, we will compare the coordinates of a vertex before and after the translation. Let's choose vertex [tex]\(A\)[/tex] and [tex]\(A'\)[/tex].

- The original coordinates of [tex]\(A\)[/tex] are [tex]\((-6, -2)\)[/tex].
- The translated coordinates of [tex]\(A'\)[/tex] are [tex]\((-10, 1)\)[/tex].

To find the translation vector [tex]\((\Delta x, \Delta y)\)[/tex], we calculate:
[tex]\[ \Delta x = x' - x = -10 - (-6) = -10 + 6 = -4 \][/tex]
[tex]\[ \Delta y = y' - y = 1 - (-2) = 1 + 2 = 3 \][/tex]
So, the translation vector is [tex]\((-4, 3)\)[/tex].

### Step 3: Identify the Translation Rule
Given the translation vector [tex]\((-4, 3)\)[/tex], we need to match this with one of the provided translation rules. The options are:
- [tex]\(T_{-4,3}^3 (x, y)\)[/tex]
- [tex]\(T_{-4,1} (x, y)\)[/tex]
- [tex]\(T_{4,-1} (x, y)\)[/tex]
- [tex]\(T_{4,-3} (x, y)\)[/tex]

The correct translation vector [tex]\((-4, 3)\)[/tex] corresponds to the rule:
[tex]\[ T_{-4,3}^3(x, y) \][/tex]

### Conclusion
The translation rule used to translate the image of rectangle [tex]\(ABCD\)[/tex] to [tex]\(A'B'C'D'\)[/tex] is:
[tex]\[ T_{-4,3}^3(x, y) \][/tex]
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for visiting IDNLearn.com. We’re here to provide accurate and reliable answers, so visit us again soon.