Question

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. i and iii b. i c. ii and iii d. ii reset next

Explanation:

Step1: Recall the properties of dilation

Dilation is a transformation that changes the size of a figure but not its shape. When a parallelogram is dilated by a scale factor \(k = 3\):

• The lengths of the sides are multiplied by the scale factor. So, if the original side lengths are \(a\) and \(b\), the new side lengths are \(3a\) and \(3b\).
• The perimeter \(P\) of a parallelogram is \(P=2(a + b)\), and the new perimeter \(P'=2(3a + 3b)=3\times2(a + b)\), so the perimeter changes.
• The area \(A\) of a parallelogram (base \(\times\) height). If the base \(b\) and height \(h\) are scaled by 3, the new area \(A'=(3b)\times(3h)=9bh\), so the area changes.
• The measure of the angles remains the same because dilation is a similarity transformation. Similarity transformations preserve angle measures.

Answer:

B. Ⅱ