To solve this problem, we need to understand how dilation affects the lengths and positions of segments in a quadrilateral. Let's go through the problem step-by-step:

Step 1: Understanding the Given Information

1. Quadrilateral ABCD:

   • ∠ABC is a right angle.
   • $AB = 4$ units.

2. Dilation Details:

   • The quadrilateral ABCD is dilated by a scale factor of 2.
   • Point $B$ is the center of dilation.
   • The resulting image is quadrilateral A'B'C'D'.

Step 2: Effect of Dilation with Center at B

When a figure is dilated with a center of dilation at a point and a scale factor $k$, each point on the figure moves along the line connecting it to the center by a distance that is $k$ times its original distance from the center.

In this problem:

• The center of dilation is point $B$.
• The scale factor is $2$.

Step 3: Analyzing the Segment $AB$ and $A'B'$

• Since $B$ is the center of dilation, point $B$ itself does not move; it remains in its original position.
• Point $A$ is 4 units away from point $B$ on line $AB$.
• Under dilation with a scale factor of 2, point $A$ moves to a new position, $A'$, such that $A'$ is 2 times the original distance from $B$ along the same line. Thus, $A'B' = 2 \times AB = 2 \times 4 = 8$ units.

Step 4: Determining the Line of A'B'

Since dilation occurs along the line connecting point $A$ to point $B$, the line $A'B'$ will lie on the same line as $AB$.

Conclusion

• A'B' is 8 units long.
• A'B' lies on the same line as AB.

Thus, the correct statement is:

Answer: B. A'B' is 8 units long and lies on the same line as AB.

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Would you like further details or have any questions? Here are some related questions:

1. How does dilation affect the angles of a geometric figure?
2. What happens to the area of a quadrilateral when it is dilated with a scale factor?
3. How can we find the coordinates of the dilated points given the original coordinates and a scale factor?
4. What is the effect of dilation on perpendicular lines in a geometric figure?
5. Can a dilation result in a figure that is congruent to the original figure?

Tip: When working with dilation problems, remember that points move along lines connecting them to the center of dilation, and the distance from the center is multiplied by the scale factor.