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.
longest side length of the original figure: units
longest side length of the final figure: units
(b) fill in the blank to make a true statement.
longest side length of the final 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 similar.
true false

Explanation:

Step1: Determine the longest side length of the original figure

By observing the original figure, the longest side length is \(2\) units.

Step2: Calculate the longest side length of the final figure

Since the scale factor is \(2\), we use the formula \( \text{Final length}=\text{Scale factor}\times\text{Original length}\). So, \(2\times2 = 4\) units.

Step3: Fill in the blank for part (b)

From the formula \( \text{Final length}=\text{Scale factor}\times\text{Original length}\), the blank is filled with the scale factor \(2\).

Step4: Analyze part (c)

A positive scale factor \(k> 1\) means \( \text{Final length}=k\times\text{Original length}\). Since \(k > 1\), the final figure is larger. So it is True.

Step5: Analyze part (d)

Dilation is a transformation that produces similar figures. So the original and final figures are similar. It is True.

Answer:

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