Questions: John says the relationship between segment BC and segment B'C' is that they are parallel. Adam says the line segments fall on the same line. Who is correct and why? John because the line segment does not pass through the center of dilation and therefore its image is parallel to the pre-image. Adam because the line segment does pass through the center of dilation and therefore its image falls on the same line as the pre-image. John because the line segment does pass through the center of dilation and therefore its image is parallel to the pre-image.

John says the relationship between segment BC and segment B'C' is that they are parallel.
Adam says the line segments fall on the same line.
Who is correct and why?
John because the line segment does not pass through the center of dilation and therefore its image is parallel to the pre-image.
Adam because the line segment does pass through the center of dilation and therefore its image falls on the same line as the pre-image.
John because the line segment does pass through the center of dilation and therefore its image is parallel to the pre-image.

Transcript text: John says the relationship between segment $B C$ and segment $B^{\prime} C^{\prime}$ 's that they are parallel. Adam says the line segments fall on the same line. Who is correct and why? John because the line segment does not pass through the center of dilation and therefore its image is parallel to the pre-image. Adam because the line segmept does pass through the center of dilation and therefore its image falls on the same line as the pre-image. John because the line segment does pass through the center of dilation and therefore its image is parallel to the pre-image.

Solution

Solution Steps

Step 1: Identify the given points and their coordinates

Points A, B, and C have coordinates (3, 5), (7, 3), and (11, 1) respectively.
Points A', B', and C' have coordinates (5, 13), (17, 7), and (29, 1) respectively.
Center of dilation P₀ has coordinates (2, 1).

Step 2: Determine the relationship between the segments BC and B'C'

Calculate the slopes of segments BC and B'C' to determine if they are parallel.
Slope of BC: $(1 - 3) / (11 - 7) = -2 / 4 = -1/2$
Slope of B'C': $(1 - 7) / (29 - 17) = -6 / 12 = -1/2$

Step 3: Determine if the segments fall on the same line

Check if the points B, C, B', and C' are collinear by verifying if they lie on the same line.
Since the slopes of BC and B'C' are equal, the segments are parallel but not necessarily collinear.