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 6 units long but lies on a different line than overline{ab}.
b. overline{ab} is 8 units long but lies on a different line than overline{ab}.
c. overline{ab} is 6 units long and lies on the same line as overline{ab}.
d. overline{ab} is 8 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 \( \overline{A'B'} \) is \( AB \times \) scale factor. \( AB = 4 \), so \( A'B' = 4 \times 2 = 8 \). Also, since center is \( B \), \( \overline{A'B'} \) lies on the same line as \( \overline{AB} \) (collinear with \( AB \) through center \( B \)).

Step2: Evaluate Options

• Option A: Length wrong (6≠8), line wrong. Eliminate.
• Option B: Length correct (8), but line wrong. Eliminate.
• Option C: Length wrong (6≠8). Eliminate.
• Option D: Length correct (8) and same line. Correct.

Answer:

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