Question

Rectangle ABCD is constructed with line EF drawn through its center. If the rectangle is dilated using a scale factor of 3 and a line is drawn through the center of the new dilated figure, what relationship will the new line have with line EF? Explain your reasoning using complete sentences.

a) The new line will be parallel to EF because the dilation preserves parallelism.
b) The new line will be perpendicular to EF because the dilation preserves perpendicularity.
c) The new line will coincide with EF because the dilation preserves collinearity.
d) The new line will have no specific relationship with EF after dilation.

Answer 1

The new line drawn through the center of a dilated rectangle coincides with the original line, preserving collinearity.

Step-by-step explanation:

The relationship between the new line drawn through the center of the dilated rectangle and the original line EF is that the new line will coincide with EF because the dilation preserves collinearity. When a geometric figure is dilated with any scale factor, the points that were collinear (lying on the same line) before the dilation remain collinear after the dilation. Since line EF was drawn through the center of rectangle ABCD, and rectangle ABCD was dilated with a scale factor of 3, the center of the dilated figure will still be on line EF. Therefore, a new line drawn through the center of the dilated figure will be the same line EF, just extended if needed to match the new dimensions.