Question

(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 = × 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 not similar. ∘ true ∘ false

Explanation:

Step1: Identify original figure's longest side

Look at the left figure (x=4 to 6, y=0 to 4). The vertical side is $4-0=4$ units, horizontal side is $6-4=2$ units, slant side: $\sqrt{(2)^2+(4)^2}=\sqrt{20}\approx4.47$? No, wait, the left figure has vertices at (4,0), (6,0), (6,4), (4,4)? No, no, the left figure is a polygon: (4,0), (5,0), (5,4), (4,4)? No, looking at the grid: original figure (left) goes from x=4 to x=6 (2 units horizontal), y=0 to y=4 (4 units vertical), and the slant side is from (4,0) to (5,4)? No, no, the longest side of original figure: the vertical side is 4 units, the slant side (from (4,0) to (6,4)) is $\sqrt{(6-4)^2+(4-0)^2}=\sqrt{4+16}=\sqrt{20}=2\sqrt{5}\approx4.47$? Wait no, the final figure (right) is from x=8 to x=12 (4 units horizontal), y=0 to y=8 (8 units vertical). Wait the slider is at 1, scale factor 2? Wait no, slider is at 1, original figure (left) longest side: let's count grid squares. Original figure: the vertical side is 4 units (from y=0 to y=4, 4 grid lines). Final figure (right): vertical side is 8 units (y=0 to y=8, 8 grid lines).

Step2: Confirm final figure's longest side

Final figure's vertical side is $8-0=8$ units, which is longer than horizontal ($12-8=4$) or slant side.

Step3: Calculate scale factor

$\text{Scale factor}=\frac{\text{Final longest side}}{\text{Original longest side}}=\frac{8}{4}=2$

Step4: Evaluate dilation statement (c)

A dilation with scale factor <1 shrinks the figure, so the statement is True.

Step5: Evaluate similarity statement (d)

Dilations produce similar figures, so "not similar" is False.

Answer:

(a)
Longest side length of the original figure: 4 units
Longest side length of the final figure: 8 units
(b)
Longest side length of the final figure = $2$ × Longest side length of the original figure
(c)
True
(d)
False