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Question 5 (Multiple Choice, Worth 5 Points)

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

A. [tex]\(N^{\prime}(1,2), M^{\prime}(-2,1), O^{\prime}(-1,3)\)[/tex]
B. [tex]\(N^{\prime}(-5,-6), M^{\prime}(-2,-5), O^{\prime}(-3,-7)\)[/tex]
C. [tex]\(N^{\prime}(-5,0), M^{\prime}(-2,-1), O^{\prime}(-3,1)\)[/tex]
D. [tex]\(N^{\prime}(9,2), M^{\prime}(6,1), O^{\prime}(7,3)\)[/tex]

Sagot :

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

Here are the given vertices of the triangle:
- [tex]\( N(-5, 2) \)[/tex]
- [tex]\( M(-2, 1) \)[/tex]
- [tex]\( O(-3, 3) \)[/tex]

For a reflection over a vertical line [tex]\( x = c \)[/tex], the coordinates [tex]\( (x', y) \)[/tex] of the reflected point are given by [tex]\( (x' = 2c - x, y = y) \)[/tex].

Let's reflect each vertex individually:

1. Vertex N:
- Original coordinates: [tex]\( N(-5, 2) \)[/tex]
- Line of reflection: [tex]\( x = -2 \)[/tex]
- Calculation: [tex]\( x' = 2(-2) - (-5) = -4 + 5 = 1 \)[/tex]
- Reflected coordinates: [tex]\( N' = (1, 2) \)[/tex]

2. Vertex M:
- Original coordinates: [tex]\( M(-2, 1) \)[/tex]
- Line of reflection: [tex]\( x = -2 \)[/tex]
- Calculation: [tex]\( x' = 2(-2) - (-2) = -4 + 2 = -2 \)[/tex]
- Reflected coordinates: [tex]\( M' = (-2, 1) \)[/tex]

3. Vertex O:
- Original coordinates: [tex]\( O(-3, 3) \)[/tex]
- Line of reflection: [tex]\( x = -2 \)[/tex]
- Calculation: [tex]\( x' = 2(-2) - (-3) = -4 + 3 = -1 \)[/tex]
- Reflected coordinates: [tex]\( O' = (-1, 3) \)[/tex]

Thus, the vertices of the reflected triangle [tex]\( N'M'O' \)[/tex] are:
- [tex]\( N'(1, 2) \)[/tex]
- [tex]\( M'(-2, 1) \)[/tex]
- [tex]\( O'(-1, 3) \)[/tex]

Comparing these results with the given multiple-choice options, the correct answer is:
[tex]\( N'(1, 2), M'(-2, 1), O'(-1, 3) \)[/tex]
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