Question

for the figure, do a dilation centered at the origin with a scale factor of 2. then answer each part. (a) fill in the blanks. shortest side length of the original figure: units shortest side length of the final figure: units (b) fill in the blank to make a true statement. shortest side length of the final figure = × shortest side length of the original figure (c) true or false? a dilation with a positive scale factor greater than 1 gives a final figure larger than the original figure. true false (d) true or false? the original figure and the final figure are not similar. true false

Explanation:

Step1: Find the shortest side length of the original figure

By observing the graph, the shortest side length of the original trapezoid is \( 2 \) units.

Step2: Calculate the shortest side length of the final figure

Since the scale factor is \( 2 \), we multiply the original side - length by the scale factor. Using the formula \( \text{Final side length}=\text{Original side length}\times\text{Scale factor} \), we get \( 2\times2 = 4 \) units.

Step3: Analyze the relationship between final and original side lengths

From the formula \( \text{Final side length}=\text{Original side length}\times\text{Scale factor} \), substituting the values we found (\(\text{Final side length} = 4\), \(\text{Original side length}=2\), \(\text{Scale factor}=2\)), we can see that \( \text{Shortest side length of the final figure}=2\times\text{Shortest side length of the original figure} \).

Step4: Determine the truth of the dilation property statement

A dilation with a positive scale factor \(k>1\) (in this case \(k = 2\)) enlarges the figure. So, the final figure is larger than the original figure. The statement “A dilation with a positive scale factor greater than \(1\) gives a final figure larger than the original figure” is True.

Step5: Determine the truth of the similarity statement

Dilation is a transformation that produces similar figures. Similar figures have the same shape but may have different sizes. So, the statement “The original figure and the final figure are not similar” is False.

Answer:

(a) Shortest side length of the original figure: \(2\) units; Shortest side length of the final figure: \(4\) units.
(b) \(2\).
(c) True.
(d) False.