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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 :

GinnyAnswer
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]
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