[tex]$\overline{Y^{\prime} Z^{\prime}}$[/tex] has endpoints located at [tex]$Y^{\prime}(0,3)$[/tex] and [tex]$Z^{\prime}(-6,3)$[/tex]. [tex]$\overline{Y Z}$[/tex] was dilated by a scale factor of 3 from the origin. Which statement describes the pre-image?

A. [tex]$\overline{Y Z}$[/tex] is located at [tex]$Y(0,9)$[/tex] and [tex]$Z(-18,9)$[/tex] and is three times the size of [tex]$\overline{Y^{\prime} Z^{\prime}}$[/tex].

B. [tex]$\overline{Y Z}$[/tex] is located at [tex]$Y(0,3)$[/tex] and [tex]$Z(-6,3)$[/tex] and is the same size as [tex]$\overline{Y^{\prime} Z^{\prime}}$[/tex].

C. [tex]$\overline{Y Z}$[/tex] is located at [tex]$Y(0,1.5)$[/tex] and [tex]$Z(-3,1.5)$[/tex] and is one-half the size of [tex]$\overline{Y^{\prime} Z^{\prime}}$[/tex].

D. [tex]$\overline{Y Z}$[/tex] is located at [tex]$Y(0,1)$[/tex] and [tex]$Z(-2,1)$[/tex] and is one-third the size of [tex]$\overline{Y^{\prime} Z^{\prime}}$[/tex].



Answer :

To solve the problem, we need to determine the correct statement describing the pre-image segment [tex]\( \overline{YZ} \)[/tex] after a dilation transformation from the segment [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex].

1. Calculate the length of [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex]:

The endpoints of [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex] are given as [tex]\( Y^{\prime}(0, 3) \)[/tex] and [tex]\( Z^{\prime}(-6, 3) \)[/tex].

The length of [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex] is calculated using the distance formula:

[tex]\[ \text{Length of } \overline{Y^{\prime} Z^{\prime}} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]

Plugging in the coordinates [tex]\( Y^{\prime}(0, 3) \)[/tex] and [tex]\( Z^{\prime}(-6, 3) \)[/tex]:

[tex]\[ \text{Length of } \overline{Y^{\prime} Z^{\prime}} = \sqrt{(-6 - 0)^2 + (3 - 3)^2} = \sqrt{(-6)^2 + 0^2} = \sqrt{36} = 6 \][/tex]

Therefore, the length of [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex] is [tex]\( 6 \)[/tex] units.

2. Determine the dilation transformation:

The problem states that [tex]\( \overline{YZ} \)[/tex] was dilated by a scale factor of [tex]\( 3 \)[/tex] from the origin.

Therefore, if the length of [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex] is [tex]\( 6 \)[/tex], then the length of [tex]\( \overline{YZ} \)[/tex] after dilation will be:

[tex]\[ \text{Length of } \overline{YZ} = 6 \times 3 = 18 \][/tex]

3. Verify the coordinates of [tex]\( \overline{YZ} \)[/tex]:

- Option 1: [tex]\( \overline{YZ} \)[/tex] is located at [tex]\( Y(0, 9) \)[/tex] and [tex]\( Z(-18, 9) \)[/tex]

Length: [tex]\( \sqrt{(-18 - 0)^2 + (9 - 9)^2} = \sqrt{(-18)^2 + 0^2} = \sqrt{324} = 18 \)[/tex]

This matches the calculated length of [tex]\( 18 \)[/tex] and also accurately reflects that [tex]\( \overline{YZ} \)[/tex] is three times the size of [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex].

- Other options:
- Option 2: [tex]\( Y(0, 3) \)[/tex] and [tex]\( Z(-6, 3) \)[/tex]

This matches [tex]\( Y^{\prime} \)[/tex] and [tex]\( Z^{\prime} \)[/tex], so it would be the same size as [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex], not three times.

- Option 3: [tex]\( Y(0, 1.5) \)[/tex] and [tex]\( Z(-3, 1.5) \)[/tex]

Length: [tex]\( \sqrt{(-3 - 0)^2 + (1.5 - 1.5)^2} = 3 \)[/tex], which is one-half, not three times.

- Option 4: [tex]\( Y(0, 1) \)[/tex] and [tex]\( Z(-2, 1) \)[/tex]

Length: [tex]\( \sqrt{(-2 - 0)^2 + (1 - 1)^2} = 2 \)[/tex], which is one-third, not three times.

Therefore, the correct statement is:

[tex]\[ \overline{YZ} \text{ is located at } Y(0, 9) \text{ and } Z(-18, 9) \text{ and is three times the size of } \overline{Y^{\prime} Z^{\prime}} \][/tex]

So the correct answer is option 1.