Question

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 abcd. which statement is true? a. (overline{ab}) is 8 units long and lies on the same line as (overline{ab}) b. (overline{ab}) is 6 units long but lies on a different line than (overline{ab}) c. (overline{ab}) is 8 units long but lies on a different line than (overline{ab}) d. (overline{ab}) is 6 units long and lies on the same line as (overline{ab})

Explanation:

Step1: Recall Dilation Properties

Dilation with center \( B \) and scale factor \( 2 \): length of \( \overline{A'B'} = \text{scale factor} \times \overline{AB} \). Given \( AB = 4 \), so \( A'B' = 2 \times 4 = 8 \). Also, since dilation from center \( B \), \( \overline{A'B'} \) lies on the same line as \( \overline{AB} \) (as \( B \) is the center, the image of \( A \) ( \( A' \)) lies along the line through \( A \) and \( B \)).

Step2: Evaluate Options

• Option A: \( A'B' = 8 \) (from \( 2\times4 \)) and same line as \( AB \) (due to center \( B \)) – matches.
• Option B: Length is 6 (incorrect, should be 8) – eliminate.
• Option C: Length 8 but different line (incorrect, same line) – eliminate.
• Option D: Length 6 (incorrect) – eliminate.

Answer:

A. \(\overline{A'B'}\) is 8 units long and lies on the same line as \(\overline{AB}\)