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If point P (2,-8) is reflected across the line y = -2, what are the coordinates of its reflection image? Write your answer as a coordinate point. (2 points)
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The vertices of a triangle are P(–7, –4), Q(–7, –8), and R(3, –3). Name the vertices of the image reflected across the line x axis.(3 points)
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How many lines of symmetry are found in a regular polygon with 70 sides?(1 point)
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A dilation has center (0, 0, 0). Find the image of the point (–2, –3, 0) for the scale factor of 3.
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The endpoints of AB¯¯¯¯¯¯¯¯ are A (8,4) and B(6,-2). The endpoints of its image after a dilation are A' (5,2.5) and B'(3.75,-1.25). Find the scale factor. Please enter your scale factor as a whole number or a fraction. No Decimals(3 points)
Answered by GPT 3.5
The line of reflection is y = -2.

To find the reflected point, we need to find the distance between point P and the line y = -2. This distance will be the same for both the original point and its reflection.

The distance between point P (2, -8) and the line y = -2 is 6 units (from -8 to -2).

Since the reflection will be the same distance from the line but on the opposite side, the reflected point will be 6 units above the line y = -2.

Therefore, the coordinates of the reflected point will be (2, 2).

Therefore, the coordinates of the reflection image are (-2, 2).
Answered by GPT 3.5
To reflect a point across the x-axis, we change the sign of the y-coordinate while keeping the x-coordinate the same.

For point P(–7, –4), the image point P' after reflecting across the x-axis will have the coordinates (–7, 4).

For point Q(–7, –8), the image point Q' after reflecting across the x-axis will have the coordinates (–7, 8).

For point R(3, –3), the image point R' after reflecting across the x-axis will have the coordinates (3, 3).

Therefore, the vertices of the new triangle after reflecting across the x-axis are P'(–7, 4), Q'(–7, 8), and R'(3, 3).
Answered by GPT 3.5
A regular polygon with n sides has n lines of symmetry.

Therefore, a regular polygon with 70 sides will have 70 lines of symmetry.
Answered by GPT 3.5
To find the image of the point (-2, -3, 0) after the dilation with center (0, 0, 0) and a scale factor of 3, we simply multiply the coordinates of the point by the scale factor.

Image of the point = (scale factor) * (original coordinates)

Image of the point = 3 * (-2, -3, 0)

Image of the point = (-6, -9, 0)

Therefore, the image of the point (-2, -3, 0) after dilation with a scale factor of 3 and center (0, 0, 0) is (-6, -9, 0).
Answered by GPT 3.5
To find the scale factor of the dilation, we can compare the change in coordinates from the original to the image endpoints.

Let's first find the change in x and y coordinates for point A:
Change in x-coordinate for A: 5 - 8 = -3
Change in y-coordinate for A: 2.5 - 4 = -1.5

Let's now find the change in x and y coordinates for point B:
Change in x-coordinate for B: 3.75 - 6 = -2.25
Change in y-coordinate for B: -1.25 - (-2) = 0.75

Now, we can calculate the dilation scale factor:
Scale factor = Change in x-coordinate (A) / Change in x-coordinate (B)
Scale factor = -3 / -2.25
Scale factor = 4/3

Therefore, the scale factor for the dilation is 4/3.