Question

quadrilateral ( abcd ) is the result of dilating quadrilateral ( abcd ) about point ( p ) by a scale factor of ( \frac{5}{3} ). determine whether each claim about the properties of ( abcd ) and ( abcd ) is true or false. ( bc ) and ( bc ) are on the same line. the lengths of ( cd ) and ( cd ) are equal.

Explanation:

Step1: Understand the property of dilation

Dilation is a transformation that changes the size of a figure but not its shape. Corresponding points of the original figure and the dilated figure lie on the same line passing through the center of dilation.
Since \(A'B'C'D'\) is the dilation of \(ABCD\) about point \(P\), \(BC\) and \(B'C'\) lie on the same line (because dilation preserves collinearity of points with respect to the center of dilation).

Step2: Analyze the lengths of corresponding sides

The scale factor of dilation is \(k = \frac{5}{3}\). If the length of a side \(l\) in the original figure \(ABCD\) (e.g., \(CD\)) and the length of the corresponding side \(l'\) in the dilated figure \(A'B'C'D'\) (e.g., \(C'D'\)), then the relationship is \(l'=k\times l\). Since \(k=\frac{5}{3}
eq1\), the lengths of \(CD\) and \(C'D'\) are not equal.

Answer:

• \(BC\) and \(B'C'\) are on the same line: True
• The lengths of \(CD\) and \(C'D'\) are equal: False