Answered

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Solve for [tex]\( x \)[/tex].

[tex]\[ 3x = 6x - 2 \][/tex]

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(02.01 LC)

[tex]\[ \triangle DOG \][/tex] has coordinates [tex]\( D(3,2), O(2,-4), \)[/tex] and [tex]\( G(-1,-1) \)[/tex]. A translation maps point [tex]\( D \)[/tex] to [tex]\( D^{\prime}(2,4) \)[/tex]. Find the coordinates of [tex]\( O^{\prime} \)[/tex] and [tex]\( G^{\prime} \)[/tex] under this translation.

A. [tex]\( O^{\prime}(-1,1) ; G^{\prime}(1,-2) \)[/tex]

B. [tex]\( O^{\prime}(0,-3) ; G^{\prime}(-3,0) \)[/tex]

C. [tex]\( O^{\prime}(1,-2) ; G^{\prime}(-2,1) \)[/tex]

D. [tex]\( O^{\prime}(3,0) ; G^{\prime}(-1,-1) \)[/tex]

Sagot :

GinnyAnswer
To find the coordinates of [tex]\(O'\)[/tex] and [tex]\(G'\)[/tex] under the given translation, follow these steps:

1. Identify the translation vector:
- We need to translate point [tex]\(D(3,2)\)[/tex] to [tex]\(D'(2,4)\)[/tex].
- Calculate the translation vector by subtracting the coordinates of [tex]\(D\)[/tex] from [tex]\(D'\)[/tex]:
[tex]\[ \text{Translation vector} = (2 - 3, 4 - 2) = (-1, 2) \][/tex]

2. Apply the translation vector to point [tex]\(O\)[/tex]:
- The initial coordinates of [tex]\(O\)[/tex] are [tex]\(O(2, -4)\)[/tex].
- Add the translation vector [tex]\((-1, 2)\)[/tex] to the coordinates of [tex]\(O\)[/tex]:
[tex]\[ O' = (2 + (-1), -4 + 2) = (1, -2) \][/tex]

3. Apply the translation vector to point [tex]\(G\)[/tex]:
- The initial coordinates of [tex]\(G\)[/tex] are [tex]\(G(-1, -1)\)[/tex].
- Add the translation vector [tex]\((-1, 2)\)[/tex] to the coordinates of [tex]\(G\)[/tex]:
[tex]\[ G' = (-1 + (-1), -1 + 2) = (-2, 1) \][/tex]

Hence, the coordinates of [tex]\(O'\)[/tex] and [tex]\(G'\)[/tex] after the translation are [tex]\(O'(1, -2)\)[/tex] and [tex]\(G'(-2, 1)\)[/tex] respectively.

Therefore, the correct answer is:
[tex]\[ O'(1, -2); G'(-2, 1) \][/tex]
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