Question

Line AB is dilated by a scale factor of 2 centered at point A. Evan thinks that the dilation of AB will result in a line parallel to AB, not passing through points A or B. Nathan thinks that the dilation of AB will result in the same line, AB. Who is correct? Explain why

Answer 1

Nathan is correct; the dilation of line AB by a scale factor of 2 centered at point A will result in the same line, not a parallel line, as dilations proportionally expand the figure from the center point.

Step-by-step explanation:

In the context of geometry, when line AB is dilated by a scale factor of 2 centered at point A, the result will be a line segment, starting from A, which is twice as long as AB, but still collinear with line AB. Therefore, Nathan is correct; the dilation will result in the same line, AB, not a parallel line. Evan's assumption that the line will be parallel and not pass through points A or B is incorrect because dilations expand or contract a figure proportionally from a center point, in this case, point A. Hence, while point B will move to a new location on the line (let's call this point B'), line AB will indeed remain the same line after dilation.

Answer 2

Nathan is correct because the dilation of line AB with a scale factor of 2 centered at A results in a line segment on the same line as AB but extended further from A.

Step-by-step explanation:

When line AB is dilated with a scale factor of 2 centered at point A, the result is a new line segment that starts from point A and extends twice as far in the same direction as AB, ending at a new point, which we can call B'.

Since the dilation is centered at A, point A remains unchanged, and B' is simply twice as far from A as B was. As a result, the new line segment AB' is on the same line as AB, making Nathan correct.

Evan's belief that the line would be parallel and not pass through points A or B is incorrect because dilation maintains the direction of the line while changing the distance between points proportionally.