Answered

Polygon [tex]$ABCD$[/tex] is dilated by a scale factor of 2 with the center of dilation at the origin to create polygon [tex]$A^{\prime}B^{\prime}CD^{\prime}$[/tex].

If the endpoints of [tex]$\overline{AB}$[/tex] are located at [tex]$(0, -7)$[/tex] and [tex]$(8, 8)$[/tex], what is the length of [tex]$\overline{A^{\prime}B^{\prime}}$[/tex]?

Use the distance formula to help you decide:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]



Answer :

GinnyAnswer
Certainly! To determine the length of \( \overline{A'B'} \), we first need to calculate the original length of \( \overline{AB} \). Here are the steps:

1. Identify the coordinates of points \(A\) and \(B\):
- Point \(A\) has coordinates \((0, -7)\).
- Point \(B\) has coordinates \((8, 8)\).

2. Apply the distance formula:
The distance formula is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]

3. Substitute the coordinates of \(A\) and \(B\) into the distance formula:
Here, \((x_1, y_1) = (0, -7)\) and \((x_2, y_2) = (8, 8)\).

[tex]\[ d = \sqrt{(8 - 0)^2 + (8 - (-7))^2} \][/tex]

4. Simplify the expression:
[tex]\[ d = \sqrt{(8)^2 + (8 + 7)^2} \][/tex]
[tex]\[ d = \sqrt{64 + 225} \][/tex]
[tex]\[ d = \sqrt{289} \][/tex]
[tex]\[ d = 17 \][/tex]

So, the length of \( \overline{AB} \) is \( 17 \).

5. Apply the scale factor of 2:
Since the polygon is dilated by a scale factor of 2, the length of each side of the polygon will also be scaled by a factor of 2.

Therefore, the length of \( \overline{A'B'} \) will be:
[tex]\[ 2 \times 17 = 34 \][/tex]

So, the length of [tex]\( \overline{A'B'} \)[/tex] after the dilation is [tex]\( 34 \)[/tex].

Other Questions