Question

consider figure abcd.
figure abcd is the image of figure abcd under a dilation with center a and scale factor 1/3.
which statements about figure abcd are true?
select three that apply.
a (overline{ab}) is parallel to (overline{dc})
b (overline{ad}) is parallel to (overline{bc})
c the measure of (angle a) is (33^{circ})
d the measure of (angle d) is (81^{circ})
e the length of (overline{ab}) equals the length of (overline{ab})

Explanation:

Step1: Properties of Dilation

Dilation is a transformation that changes the size of a figure but not its shape. Corresponding angles of the original figure and its dilated image are equal. Corresponding sides of the original figure and its dilated image are parallel (if the center of dilation is not on the side) and in proportion to the scale factor.

Step2: Analyze Option A

Since \(AB\parallel DC\) in the original figure \(ABCD\), after dilation with center \(A\), \(A'B'\parallel AB\) (because dilation from center \(A\), \(A\) is a fixed - point for \(A'B'\) and \(AB\)). Also, \(D'C'\parallel DC\). Since \(AB\parallel DC\), by the transitive property of parallel lines (\(a\parallel b\) and \(b\parallel c\) implies \(a\parallel c\)), \(A'B'\parallel D'C'\).

Step3: Analyze Option B

In the original figure \(ABCD\), \(AD\) and \(BC\) are not parallel. Dilation does not change the non - parallel relationship. If we assume \(A'D'\) and \(B'C'\) were parallel, but the original non - parallel relationship of \(AD\) and \(BC\) (since \(ABCD\) is a trapezoid with \(AB\parallel DC\) only) is preserved under dilation (as dilation is a similarity transformation which preserves the nature of line - line relationships: parallel or non - parallel). So \(A'D'\) is not parallel to \(B'C'\).

Step4: Analyze Option C

Angles are preserved under dilation. The measure of \(\angle A\) in figure \(ABCD\) is \(99^{\circ}\). Since \(\angle A'\) and \(\angle A\) are corresponding angles (\(\angle A'\) is the image of \(\angle A\) under dilation), \(m\angle A'=m\angle A = 99^{\circ}
eq33^{\circ}\).

Step5: Analyze Option D

Angles are preserved under dilation. The measure of \(\angle D\) in figure \(ABCD\) is \(81^{\circ}\). Since \(\angle D'\) and \(\angle D\) are corresponding angles (\(\angle D'\) is the image of \(\angle D\) under dilation), \(m\angle D'=m\angle D = 81^{\circ}\).

Step6: Analyze Option E

The length of a side in the dilated figure \(l'\) and the length of the corresponding side in the original figure \(l\) are related by the formula \(l'=k\times l\), where \(k\) is the scale factor. Here \(k = \frac{1}{3}\), so \(l(A'B')=\frac{1}{3}l(AB)
eq l(AB)\).

Answer:

A. \(\overline{A'B'}\) is parallel to \(\overline{D'C'}\), D. The measure of \(\angle D'\) is \(81^{\circ}\)