Question

(a) fill in the blanks.
longest side length of the original figure: 16 units
longest side length of the final figure: 4 units
(b) fill in the blank to make a true statement.
longest side length of the final figure = × longest 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:

Part (a)

Step1: Analyze the given data

The original longest side is 16 units, and from the problem's context (likely a dilation), the final longest side is given as 4 units (from the filled - in part, but we can also calculate it as we'll see in part (b)).

Step2: Confirm the value

Since the problem's blank for the final figure's longest side is filled with 4 (and we can also derive it from the scale factor later), the longest side length of the final figure is 4 units.

Step1: Recall the dilation formula

The formula for dilation is \( \text{Final Length}=\text{Scale Factor}\times\text{Original Length} \). Let the scale factor be \( k \), the original length \( L_o = 16 \) and the final length \( L_f=4 \).

Step2: Solve for the scale factor

We know that \( L_f=k\times L_o \), so \( k=\frac{L_f}{L_o}=\frac{4}{16}=\frac{1}{4} \). So the longest side length of the final figure \(=\frac{1}{4}\times \) longest side length of the original figure.

Step1: Recall the definition of dilation

Dilation is a transformation that changes the size of a figure. If the scale factor \( k \) is positive and \( 0 < k< 1 \), the image (final figure) is a reduction of the original figure, meaning it is smaller than the original figure.

Step2: Evaluate the statement

Since a dilation with a positive scale factor less than 1 makes the figure smaller, the statement is True.

Answer:

4

Part (b)