Question

1. a line segment is dilated by a scale factor of 2 centered at a point not on the line segment. which statement regarding the relationship between the given line segment and its image is true?

a. the line segments are perpendicular, and the image is one - half of the length of the given line segment.
b. the line segments are perpendicular, and the image is twice the length of the given line segment.
c. the line segments are parallel, and the image is twice the length of the given line segment.
d. the line segments are parallel, and the image is one - half of the length of the given line segment.

1. line n is represented by the equation 3x + 4y = 20. determine and state the equation of line p, the image of line n, after a dilation of scale factor \\(\frac{1}{2}\\) centered at the point (4, 2).

explain your answer.

1. a three - inch line segment is dilated by a scale factor of 6 and centered at its midpoint. what is the length of its image?

a. 9 inches
b. 2 inches
c. 15 inches
d. 18 inches

1. which sequence of transformations will map \\(\triangle abc\\) onto \\(\triangle abc\\)?

a. reflection and translation
b. rotation and reflection
c. translation and dilation
d. dilation and rotation

Explanation:

Question 13

Step1: Recall Dilation Properties

Dilation preserves the parallelism of line segments (or lines) and scales the length by the scale factor. The scale factor here is 2, so the image length is \( 2 \times \) original length. Also, dilation doesn't make segments perpendicular (perpendicular is for rotations/reflections in some cases, not dilation).

Step2: Analyze Options

• Option A: Image length is half, wrong (scale factor 2 means double). Also, not perpendicular. Eliminate.
• Option B: Not perpendicular (dilation preserves parallelism, not perpendicularity). Eliminate.
• Option C: Segments are parallel (dilation preserves parallelism) and image is twice the length (scale factor 2). Correct.
• Option D: Image length is half, wrong. Eliminate.

Step1: Recall Dilation of a Line

When a line is dilated with a scale factor \( k \) centered at a point, if the center lies on the line, the image is the same line. First, check if \( (4,2) \) lies on \( 3x + 4y = 20 \). Substitute \( x = 4 \), \( y = 2 \): \( 3(4) + 4(2) = 12 + 8 = 20 \). So the center is on the line.

Step2: Dilation of Line Through Center

If a line is dilated with a center on the line, the image is the same line (since all points on the line, when dilated, remain on the line). So the equation of line \( p \) is the same as line \( n \), \( 3x + 4y = 20 \).

Step1: Recall Dilation Length Formula

When a segment is dilated by scale factor \( k \), the image length is \( k \times \) original length. Original length is 3 inches, scale factor 6.

Step2: Calculate Image Length

Image length \( = 3 \times 6 = 18 \) inches.

Answer:

C. The line segments are parallel, and the image is twice the length of the given line segment.

Question 14