In quadrilateral ABCD, ∠ABC is a right angle and AB = 4 units. Quadrilateral ABCD is dilated by a scale factor of 2 with point B as the center of dilation, resulting in the image, quadrilateral A'B'C'D'. Which statement is true?
 A. 
A'B' is 8 units long but lies on a different line than AB.
 B. 
A'B' is 8 units long and lies on the same line as AB.
 C. 
A'B' is 6 units long and lies on the same line as AB.
 D. 
A'B' is 6 units long but lies on a different line than AB.


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.

In quadrilateral ABCD, ∠ABC is a right angle and AB = 4 units. Quadrilateral ABCD is dilated by a scale factor of 2 with point B as the center of dilation, resulting in the image, quadrilateral A'B'C'D'. Which statement is true? A. A'B' is 8 units long but lies on a different line than AB. B. A'B' is 8 units long and lies on the same line as AB. C. A'B' is 6 units long and lies on the same line as AB. D. A'B' is 6 units long but lies on a different line than AB.

Explore the effects of dilation on quadrilateral ABCD with a scale factor of 2, where ∠ABC is a right angle and AB = 4 units. Learn how dilation affects the length and position of segments with point B as the center. Suitable for Grades 8-10.

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