Answered

Triangle EFG has vertices [tex]\(E(-3,4)\)[/tex], [tex]\(F(-5,-1)\)[/tex], and [tex]\(G(1,1)\)[/tex]. The triangle is translated so that the coordinates of the image are [tex]\(E^{\prime}(-1,0)\)[/tex], [tex]\(F^{\prime}(-3,-5)\)[/tex], and [tex]\(G^{\prime}(3,-3)\)[/tex].

Which rule was used to translate the image?

A. [tex]\(T_{4,-4}(x, y)\)[/tex]
B. [tex]\(T_{-4,-4}(x, y)\)[/tex]
C. [tex]\(T_{2,-4}(x, y)\)[/tex]
D. [tex]\(T_{-2,-4}(x, y)\)[/tex]



Answer :

GinnyAnswer
To determine the rule used to translate the image of triangle [tex]\( EFG \)[/tex] to triangle [tex]\( E'F'G' \)[/tex], we need to identify the translation vector that maps the original vertices to their corresponding image vertices.

Let's determine the translation vector for each vertex:

1. For vertex [tex]\( E \)[/tex]:

- Original coordinates of [tex]\( E \)[/tex]: [tex]\( (-3, 4) \)[/tex]
- New coordinates of [tex]\( E' \)[/tex]: [tex]\( (-1, 0) \)[/tex]
- Translation vector: [tex]\( (E'_x - E_x, E'_y - E_y) = (-1 - (-3), 0 - 4) = (-1 + 3, 0 - 4) = (2, -4) \)[/tex]

2. For vertex [tex]\( F \)[/tex]:

- Original coordinates of [tex]\( F \)[/tex]: [tex]\( (-5, -1) \)[/tex]
- New coordinates of [tex]\( F' \)[/tex]: [tex]\( (-3, -5) \)[/tex]
- Translation vector: [tex]\( (F'_x - F_x, F'_y - F_y) = (-3 - (-5), -5 - (-1)) = (-3 + 5, -5 + 1) = (2, -4) \)[/tex]

3. For vertex [tex]\( G \)[/tex]:

- Original coordinates of [tex]\( G \)[/tex]: [tex]\( (1, 1) \)[/tex]
- New coordinates of [tex]\( G' \)[/tex]: [tex]\( (3, -3) \)[/tex]
- Translation vector: [tex]\( (G'_x - G_x, G'_y - G_y) = (3 - 1, -3 - 1) = (2, -4) \)[/tex]

All translations vectors are the same: [tex]\( (2, -4) \)[/tex].

Hence, the translation rule used to translate the triangle [tex]\( EFG \)[/tex] to [tex]\( E'F'G' \)[/tex] is [tex]\( T_{2, -4}(x, y) \)[/tex].

The correct answer is:
[tex]\[ T_{2, -4}(x, y) \][/tex]

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