Right triangle [tex]\(LMN\)[/tex] has vertices [tex]\(L(7,-3)\)[/tex], [tex]\(M(7,-8)\)[/tex], and [tex]\(N(10,-8)\)[/tex]. The triangle is translated so that the coordinates of [tex]\(L'\)[/tex] are [tex]\((-1, 8)\)[/tex].

Which rule was used to translate the image?

A. [tex]\((x, y) \rightarrow (x + 6, y - 5)\)[/tex]
B. [tex]\((x, y) \rightarrow (x - 6, y + 5)\)[/tex]
C. [tex]\((x, y) \rightarrow (x + 8, y - 11)\)[/tex]
D. [tex]\((x, y) \rightarrow (x - 8, y + 11)\)[/tex]



Answer :

To determine the rule used to translate the triangle, follow these steps:

1. Identify the original coordinates of point [tex]\( L \)[/tex] and its translated coordinates [tex]\( L' \)[/tex]:
- [tex]\( L(7,-3) \)[/tex]
- [tex]\( L'(-1,8) \)[/tex]

2. Calculate the translation vector. This vector represents how much to move in the [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-directions to get from [tex]\( L \)[/tex] to [tex]\( L' \)[/tex].
- The translation in the [tex]\( x \)[/tex]-direction is given by the difference in the [tex]\( x \)[/tex]-coordinates: [tex]\( -1 - 7 = -8 \)[/tex].
- The translation in the [tex]\( y \)[/tex]-direction is given by the difference in the [tex]\( y \)[/tex]-coordinates: [tex]\( 8 - (-3) = 8 + 3 = 11 \)[/tex].

3. Thus, the translation vector is [tex]\( (-8, 11) \)[/tex].

4. Translate this vector into the form of a translation rule. The rule expresses how to move any point [tex]\( (x, y) \)[/tex] to its new position:
- The translation rule derived from the vector [tex]\( (-8, 11) \)[/tex] is [tex]\( (x, y) \rightarrow (x-8, y+11) \)[/tex].

5. Compare the derived translation rule with the given choices:
- [tex]\( (x, y) \rightarrow (x+6, y-5) \)[/tex]
- [tex]\( (x, y) \rightarrow (x-6, y+5) \)[/tex]
- [tex]\( (x, y) \rightarrow (x+8, y-11) \)[/tex]
- [tex]\( (x, y) \rightarrow (x-8, y+11) \)[/tex]

6. The correct match is:
- [tex]\( (x, y) \rightarrow (x-8, y+11) \)[/tex]

Therefore, the rule used to translate triangle [tex]\( LMN \)[/tex] so that [tex]\( L \)[/tex] translates to [tex]\( L' \)[/tex] is:
[tex]\[ (x, y) \rightarrow (x-8, y+11) \][/tex]

The correct answer is:
[tex]\[ 4 \][/tex]