Question

for the figure, do a dilation centered at the origin with a scale factor of 1/2. 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 less than 1 gives a final figure smaller than the original figure.
true
false
(d) true or false?
the original figure and the final figure are similar.
true
false

Explanation:

Step1: Recall dilation property

When dilating a figure centered at the origin with a scale factor \(k\), the lengths of the sides of the new - figure are related to the lengths of the sides of the original figure by the formula \(l_{new}=k\times l_{old}\). Here \(k = \frac{1}{2}\).

Step2: Answer part (a)

Let the length of the shortest side of the original figure be \(x\) units. The length of the shortest side of the final figure \(l_{new}\) is given by \(l_{new}=\frac{1}{2}\times x\). If we assume the length of the shortest side of the original figure is \(4\) units (by measuring or observing, not given in the problem - but for illustration), then the length of the shortest side of the original figure: \(4\) units, and the length of the shortest side of the final figure: \(\frac{1}{2}\times4 = 2\) units.

Step3: Answer part (b)

The relationship for the shortest - side lengths is Shortest side length of the final figure=\(\frac{1}{2}\times\)Shortest side length of the original figure.

Step4: Answer part (c)

A dilation with a positive scale factor \(k\lt1\) (in this case \(k = \frac{1}{2}\)) gives a final figure smaller than the original figure. This is true because if \(l_{new}=k\times l_{old}\) and \(0\lt k\lt1\), then \(l_{new}\lt l_{old}\).

Step5: Answer part (d)

When a figure is dilated, the original figure and the final figure are similar. This is a fundamental property of dilation. Corresponding angles are equal and corresponding sides are in proportion.

Answer:

(a) Let's assume the shortest side length of the original figure: \(4\) units, shortest side length of the final figure: \(2\) units
(b) Shortest side length of the final figure=\(\frac{1}{2}\times\)Shortest side length of the original figure
(c) True
(d) True