Line AB is on the coordinate plane and does not pass through the origin. Line AB is dilated with a scale factor of 3 and a center of dilation at the origin to create line A'B'. Describe the effect of the dilation on line AB. In particular, make sure to describe the relative location and the size of line A'B' compared to line AB. If line AB was dilated with a scale factor of 1/3, how would your answer change?
When line AB is dilated with a scale factor of 3 and a center of dilation at the origin, line A'B' will be three times the length of line AB and will be located at the same relative position as line AB, meaning it will have the same slope and direction.
If line AB was dilated with a scale factor of 1/3, line A'B' would be one-third the length of line AB and would still be located at the same relative position as line AB. The scale factor of 1/3 indicates that all distances on line AB would be contracted to one-third of their original length when creating line A'B'.
The dilation with a scale factor of 3 and a center of dilation at the origin stretches or expands line AB. The relative location of line A'B' compared to line AB will depend on the specific position of line AB on the coordinate plane.
If line AB lies entirely in a single quadrant, line A'B' will also lie in the same quadrant, maintaining the same general direction but being three times as long. The size of line A'B' will be three times longer than line AB.
If line AB passes through multiple quadrants, the relative location of line A'B' will still maintain the same general direction, but it will be three times as long, extending into the corresponding quadrants.
If line AB was dilated with a scale factor of 1/3, it would have the opposite effect. Line AB would shrink or contract, and line A'B' would be one-third the size of line AB. The relative location would remain the same, maintaining the same general direction.
To understand the effect of dilation, let's start by explaining what dilation is. Dilation is a transformation that resizes a figure without changing its shape. It uses a scale factor, which can be either greater than 1 (enlargement) or between 0 and 1 (reduction).
In this case, line AB is dilated with a scale factor of 3 and a center of dilation at the origin, resulting in line A'B'.
When a line is dilated with a scale factor greater than 1, such as 3, the effect is an enlargement. The new line, A'B', will be three times as long as line AB, but will retain the same slope and direction. The location of A'B' will depend on the position of point B relative to the origin.
If line AB does not pass through the origin, the effect of the dilation is that line A'B' will be located at the same angle, but will be three times farther from the origin compared to line AB. The size of line A'B' will be three times the size of line AB. In other words, line A'B' will stretch away from the origin, while still maintaining the same slope and direction as line AB.
Now, let's consider if line AB was dilated with a scale factor of 1/3. In this case, the scale factor is between 0 and 1, indicating a reduction. The effect would be that line A'B' would be one-third the length of line AB, but would retain the same slope and direction. The location of A'B' would depend on the position of point B relative to the origin.
If line AB does not pass through the origin, the effect of the dilation would be that line A'B' would be located at the same angle, but would be one-third as far from the origin compared to line AB. The size of line A'B' would be one-third the size of line AB. In other words, line A'B' would shrink towards the origin, while still maintaining the same slope and direction as line AB.
In summary, when line AB is dilated with a scale factor of 3, it results in an enlargement of line A'B', making it three times as long and located three times farther from the origin compared to line AB. When line AB is dilated with a scale factor of 1/3, line A'B' is reduced to one-third the size and located one-third as far from the origin compared to line AB.