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Question 10 (Multiple Choice, Worth 5 points)

Triangle [tex]\(NMO\)[/tex] has vertices at [tex]\(N(-5,2), M(-2,1),\)[/tex] and [tex]\(O(-3,3)\)[/tex]. Determine the vertices of image [tex]\(N^{\prime}M^{\prime}O^{\prime}\)[/tex] if the preimage is reflected over [tex]\(y = -2\)[/tex].

A. [tex]\(N^{\prime}(-3,2), M^{\prime}(0,1), O^{\prime}(-5,3)\)[/tex]
B. [tex]\(N^{\prime}(-5,0), M^{\prime}(-2,-1), O^{\prime}(-3,1)\)[/tex]
C. [tex]\(N^{\prime}(-5,1), M^{\prime}(-2,0), O^{\prime}(-3,2)\)[/tex]
D. [tex]\(N^{\prime}(-5,-6), M^{\prime}(-2,-5), O^{\prime}(-3,-7)\)[/tex]



Answer :

GinnyAnswer
To solve this problem, we need to reflect the vertices of triangle [tex]\(NMO\)[/tex] over the line [tex]\(y = -2\)[/tex].

1. Identifying the vertices of the original triangle:
- [tex]\(N(-5, 2)\)[/tex]
- [tex]\(M(-2, 1)\)[/tex]
- [tex]\(O(-3, 3)\)[/tex]

2. Reflection over the line [tex]\(y = -2\)[/tex]:
The formula for reflecting a point [tex]\((x, y)\)[/tex] over the line [tex]\(y = k\)[/tex] is [tex]\((x, 2k - y)\)[/tex]. In this case, [tex]\(k = -2\)[/tex].

3. Reflect each vertex:
- For [tex]\(N(-5, 2)\)[/tex]:
- Reflected point [tex]\(N'\)[/tex] will have coordinates [tex]\( (x, 2(-2) - y) = (-5, 2(-2) - 2) = (-5, -4 - 2) = (-5, -6)\)[/tex]

- For [tex]\(M(-2, 1)\)[/tex]:
- Reflected point [tex]\(M'\)[/tex] will have coordinates [tex]\( (x, 2(-2) - y) = (-2, 2(-2) - 1) = (-2, -4 - 1) = (-2, -5)\)[/tex]

- For [tex]\(O(-3, 3)\)[/tex]:
- Reflected point [tex]\(O'\)[/tex] will have coordinates [tex]\( (x, 2(-2) - y) = (-3, 2(-2) - 3) = (-3, -4 - 3) = (-3, -7)\)[/tex]

4. Determine the reflected vertices:
- [tex]\(N'(-5, -6)\)[/tex]
- [tex]\(M'(-2, -5)\)[/tex]
- [tex]\(O'(-3, -7)\)[/tex]

Given these calculations, the correct answer from the multiple-choice options is:
[tex]\[ N^{\prime}(-5, -6), M^{\prime}(-2, -5), O^{\prime}(-3, -7) \][/tex]

Therefore, the correct choice is:
[tex]\[ \boxed{N^{\prime}(-5, -6), M^{\prime}(-2, -5), O^{\prime}(-3, -7)} \][/tex]

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