Question

1
a parallelogram is dilated by a scale factor of three.
which of the following properties are unchanged?
i. the perimeter and area
ii. the measure of the angles
iii. the lengths of the sides
a. ii
b. ii and iii
c. i
d. i and iii
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Explanation:

Step1: Recall the properties of dilation

Dilation is a transformation that changes the size of a figure. The scale factor affects the lengths of the sides. If the scale factor is \(k = 3\), the new side - length \(s_{new}=k\times s_{old}\). So, the lengths of the sides (III) change.
The perimeter of a parallelogram \(P = 2(a + b)\) (where \(a\) and \(b\) are the side - lengths). Since \(a_{new}=3a\) and \(b_{new}=3b\), \(P_{new}=2(3a + 3b)=3\times2(a + b)=3P_{old}\). So, the perimeter (part of I) changes.
The area of a parallelogram \(A = base\times height\). If the base \(b_{new}=3b\) and the height \(h_{new}=3h\) (because of dilation), \(A_{new}=(3b)\times(3h)=9bh = 9A_{old}\). So, the area (part of I) changes.

Step2: Recall the angle - preserving property of dilation

Dilation is a similarity transformation. Similarity transformations (such as dilation) preserve the measures of angles. So, the measure of the angles (II) remains unchanged.

Answer:

A. II