Question

1. the figure shows line ab and line cd intersecting at point o. which statements about the images of lines ab and cd would be true under the dilation? select all the statements that apply.

pick up to 0 answers.
a a dilation of line ab with center d and a scale factor of 2 results in a line parallel to line ab.
b. a dilation of line ab with center c and a scale factor of 1 results in a line that is the same as line ab.
c. a dilation of line cd with center d and a scale factor of 2 results in a line that is the same as line cd.
d. a dilation of segment ab with center c and a scale factor of 2 results in a segment perpendicular to ab.
e. a dilation of segment cd with center b and a scale factor of 3 results in a segment parallel to cd and intersecting line ab.

Explanation:

Step1: Recall properties of dilation

• Dilation is a transformation that enlarges or reduces a figure. If the center of dilation is on the line, and the scale factor is \(k = 1\), the line remains unchanged. If the center of dilation is not on the line, lines are parallel (for non - zero scale factor).
• For a line \(l\) and a dilation with center \(O\) not on \(l\), the image of \(l\) is parallel to \(l\) when the scale factor \(k

eq0\). If the center of dilation \(O\) is on the line \(l\), and \(k = 1\), the line \(l\) is invariant (remains the same).

Step2: Analyze each option

• Option A:
• The center of dilation \(D\) is not on line \(AB\). A dilation of a line (not passing through the center of dilation) with a non - zero scale factor (\(k = 2\)) results in a parallel line. But when we dilate line \(AB\) with center \(D\), the image of line \(AB\) is not parallel to \(AB\) because the center of dilation is not on \(AB\), but the direction of the line changes in a non - parallel way (since the center is a point not on the original line and we are dealing with a line, not a segment). In fact, lines passing through the center of dilation (after dilation) will intersect the original line (if the center is not on the original line). So, this statement is false.
• Option B:
• The formula for dilation is \(T(x)=(x - C)\times k + C\), where \(C\) is the center of dilation and \(k\) is the scale factor. When \(k = 1\) and the center of dilation \(C\) is not on line \(AB\), \(T(x)=(x - C)\times1 + C=x\). So, a dilation of line \(AB\) with center \(C\) (not on \(AB\)) and scale factor \(k = 1\) results in the same line \(AB\). This statement is true.
• Option C:
• The center of dilation \(D\) is on line \(CD\). Using the formula \(T(x)=(x - D)\times k+D\). When \(k = 2\), for any point \(x\) on line \(CD\), \(T(x)=(x - D)\times2+D=2x-2D + D=2x - D\). But since \(x\) is on line \(CD\) (i.e., \(x=D + t\vec{v}\) for some vector \(\vec{v}\) along \(CD\) and scalar \(t\)), \(T(x)=2(D + t\vec{v})-D=D + 2t\vec{v}\), which is still on line \(CD\). So, a dilation of line \(CD\) with center \(D\) (on \(CD\)) and scale factor \(k = 2\) results in the same line \(CD\). This statement is true.
• Option D:
• Dilation is a similarity transformation. It does not change the angle between lines (except when the scale factor is \(0\)). A dilation of segment \(AB\) with center \(C\) (not on \(AB\)) and scale factor \(k = 2\) will result in a segment that is parallel (if \(C\) is not on \(AB\)) or coincident (if \(C\) is on \(AB\)) with \(AB\), not perpendicular. So, this statement is false.
• Option E:
• The center of dilation \(B\) is not on line \(CD\). A dilation of segment \(CD\) (a line segment) with center \(B\) (not on \(CD\)) and scale factor \(k = 3\) results in a segment parallel to \(CD\). Also, since line \(AB\) and line \(CD\) intersect at \(O\), and the image of \(CD\) (say \(C'D'\)) is parallel to \(CD\), and \(AB\) and \(CD\) intersect, \(AB\) and \(C'D'\) will also intersect. This statement is true.

Answer:

B. A dilation of line AB with center C and a scale factor of 1 results in a line that is the same as line AB; C. A dilation of line CD with center D and a scale factor of 2 results in a line that is the same as line CD; E. A dilation of segment CD with center B and a scale factor of 3 results in a segment parallel to CD and intersecting line AB.