 Dilations and Lines (Segments) MathBitsNotebook.com Terms of Use   Contact Person: Donna Roberts Concept 1: A dilation leaves a line passing through the center of the dilation unchanged.

What happens if we dilate an entire line?

If that line passes through the point which is the center of the dilation, nothing will change.
 Remember that the image point formed by a dilation will lie on a straight line connecting the pre-image point to the center of the dilation. The diagram above shows with a center of dilation, labeled O, located on the line. If we choose point B as our pre-image point, we know that its image after the dilation will lie on the line through O and B, which is, of course, . Since O is located on , the image of any point on will lie on .

Conclusion: The dilation of the line, with the center of dilation on the line, leaves the line unchanged (we get the same line again). The scale factor is of no importance.
 Keep in mind that this same concept will apply to "segments" in figures. When the segment (side) of a figure passes through the center of dilation, the segment (side) of the pre-image and its image will be on the same line. In the diagram at the right, with center of dilation at (0,-2) and scale factor of 2, we notice that passes through the center of dilation, (0,-2), which means that and will be on the same line, . Concept 2: A dilation takes a line NOT passing through the center of the dilation to a parallel line.

What happens if we dilate an entire line, but the center of the dilation in NOT on the line?

 The diagram at the right shows , center of dilation, O (not on ), and a scale factor of 1.5. By the definition of a dilation, we know , where O-A-A' and O-B-B' lie on straight lines (collinear). If we draw the straight line through O, A and A', we know that ∠OAB ∠OA'B' since dilations preserve angle measure. If we know that ∠OAB ∠OA'B', we know that since we have congruent corresponding angles. (Theorem: If two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel.) Conclusion: A dilation takes a line NOT passing through the center of the dilation to a parallel line. It is important to keep in mind that dilations also create parallel "segments" when dealing with figures. When a figure is dilated, a segment (side) of the pre-image that does not pass through the center of dilation will be parallel to its image. In the diagram at the right, with center of dilation at (-4,-9) and scale factor of 2, we have . Concept 3: The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

 If the scale factor is greater than 1 (k > 1), the image will be larger than the pre-image, making the segments (sides) of the image longer than the corresponding sides of its pre-image (an enlargement). A'C' > AC;      A'B' > AB;      B'C' > BC If the scale factor is 2, we will have A'C' = 2AC;     A'B' = 2AB;     B'C' = 2BC Also, notice that the pre-image will be between the center of dilation and the image. Note: If the absolute value of the scale factor is greater than 1, the image will be larger than the pre-image. (When k < -1, the image is larger, with a 180º rotation. The negative symbol indicates direction.) If the scale factor is between 0 and 1 (0 < k < 1), the image will be smaller than the pre-image, making the segments (sides) of the image smaller than the corresponding sides of its pre-image (a reduction). D'E' < DE;      D'F' < DF;      F'E' < FE If the scale factor is ½, we will have D'E' = ½DE;    D'F' = ½DF;    F'E' = ½FE Also notice that the image will be between the center of dilation and the pre-image. Note: If the absolute value of the scale factor is between 0 and 1, the image will be smaller than the pre-image. (When -1< k < 0, the image is smaller, with a 180º rotation. The negative symbol indicates direction.) If k = 1 (or k = -1), the image and pre-image are the same size (congruent). Given: is 10 inches long.
If is dilated to form with a scale factor of 0.8, find the length of .

Solution:
We know that the ratio of the length of the image to the length of the pre-image is equal to the scale factor.

Since we know , we can substitute AB = 10, and solve for our answer.  