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Question

A coordinate plane with four quadrants shows an x axis capped with arrows and ranging from negative 11 to 10 in 1 unit increments and a y axis capped with arrows and ranging from 12 to negative 12 in 1 unit increments. Points upper A, upper B, upper A prime, and upper B prime are plotted. A coordinate plane with four quadrants shows an x axis capped with arrows and ranging from negative 11 to 10 in 1 unit increments and a y axis capped with arrows and ranging from 12 to negative 12 in 1 unit increments. Point upper A is plotted as left parenthesis negative 3 comma 4 right parenthesis and point upper B is plotted as left parenthesis 2 comma negative 4 right parenthesis. The points are joined by a line that extends beyond the points and is capped with arrows at both ends. Point upper A prime is plotted as left parenthesis negative 6 comma 8 right parenthesis and point upper B prime is plotted as left parenthesis 4 comma negative 8 right parenthesis. The points are joined by a line that extends beyond the points and is capped with arrows at both ends.

Determine the scale factor used in the dilation of line AB
, with a center point of dilation at the origin.

(1 point)
The scale factor is
.

Answer 1

To find the scale factor of the dilation of line segment \( AB \) (from points \( A \) and \( B \) to points \( A' \) and \( B' \)), we can use the coordinates of the points before and after the dilation.

1. Identify the coordinates of the points:

   • \( A(-3, 4) \)
   • \( B(2, -4) \)
   • \( A'(-6, 8) \)
   • \( B'(4, -8) \)

2. Calculate the distance from the origin (the center of dilation) to each point before and after the dilation.

   • For point \( A \): \[ \text{Distance from origin} = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

   • For point \( B \): \[ \text{Distance from origin} = \sqrt{2^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \]

   • For point \( A' \): \[ \text{Distance from origin} = \sqrt{(-6)^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]

   • For point \( B' \): \[ \text{Distance from origin} = \sqrt{4^2 + (-8)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5} \]

3. Now, find the scale factor using the distances.

   • For point \( A \) to \( A' \): \[ \text{Scale factor} = \frac{\text{Distance to } A'}{\text{Distance to } A} = \frac{10}{5} = 2 \]

   • For point \( B \) to \( B' \): \[ \text{Scale factor} = \frac{\text{Distance to } B'}{\text{Distance to } B} = \frac{4\sqrt{5}}{2\sqrt{5}} = 2 \]

Since both calculations yield the same scale factor, we can conclude that the scale factor used in the dilation of line \( AB \) is

\[ \boxed{2} \]