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2. A quadrilateral has the following vertices:
[tex]$
\begin{array}{l}
A(0,4) \\
B(8,4) \\
C(4,-8) \\
D(-4,-8)
\end{array}
$[/tex]

If the scale factor is [tex]$\frac{1}{4}$[/tex], which of the following shows the length of segment [tex]$A^{\prime} B^{\prime}$[/tex]?

A. 4
B. 2
C. 1
D. 8


Sagot :

To find the length of segment [tex]\(A'B'\)[/tex] after scaling down the quadrilateral with a scale factor of [tex]\(\frac{1}{4}\)[/tex], we need to follow these steps:

1. Identify the coordinates of points A and B:
- [tex]\(A(0,4)\)[/tex]
- [tex]\(B(8,4)\)[/tex]

2. Calculate the original length of segment [tex]\(AB\)[/tex]:
- Using the distance formula for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], the distance [tex]\(d\)[/tex] is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
- Plugging in the coordinates of [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[ AB = \sqrt{(8 - 0)^2 + (4 - 4)^2} \][/tex]
[tex]\[ AB = \sqrt{8^2 + 0^2} \][/tex]
[tex]\[ AB = \sqrt{64} \][/tex]
[tex]\[ AB = 8 \][/tex]

3. Apply the scale factor [tex]\(\frac{1}{4}\)[/tex]:
- The length of [tex]\(A'B'\)[/tex] after applying the scale factor is:
[tex]\[ A'B' = AB \times \frac{1}{4} \][/tex]
- Substituting the length of [tex]\(AB\)[/tex]:
[tex]\[ A'B' = 8 \times \frac{1}{4} \][/tex]
[tex]\[ A'B' = 2 \][/tex]

Thus, the length of segment [tex]\(A'B'\)[/tex] after scaling is [tex]\(2\)[/tex].

Therefore, the correct answer is:

[tex]\[ \boxed{2} \][/tex]
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