Question

select the correct answer.
in the figure, $\triangle abc$ is congruent to $\triangle adc$. if the square $abcd$ is dilated by a factor of 2 to form $abcd$, what is the ratio of the area of $abcd$ to the area of $abcd$?
a. $2:1$
b. $3:1$
c. $4:1$
d. $5:1$

Explanation:

Step1: Recall dilation area ratio

When a figure is dilated by a scale factor \( k \), the ratio of the areas of the original figure to the dilated figure is \( 1:k^2 \). Here, the dilation factor \( k = 2 \).

Step2: Calculate area ratio

For the square \( ABCD \) and its dilated image \( A'B'C'D' \), the ratio of the area of \( ABCD \) to \( A'B'C'D' \) is \( 1:2^2=1:4 \), so the ratio of \( A'B'C'D' \) to \( ABCD \) is \( 4:1 \)? Wait, no—wait, the question is "the ratio of the area of \( ABCD \) to the area of \( A'B'C'D' \)"? Wait, no, let's re - read. Wait, the square \( ABCD \) is dilated by a factor of 2 to form \( A'B'C'D' \). So original area \( A \), dilated area \( A'=k^{2}A \), where \( k = 2 \), so \( A' = 4A \). So the ratio of \( ABCD \) (original) to \( A'B'C'D' \) (dilated) is \( A:A'=1:4 \)? Wait, no, the question is "what is the ratio of the area of \( A'B'C'D' \) to the area of \( ABCD \)"? Wait, the options are A. 2:1, B. 3:1, C. 4:1, D. 5:1. Wait, when you dilate a figure by scale factor \( k \), the area scales by \( k^{2} \). So if the scale factor is 2, the area of the dilated figure ( \( A'B'C'D' \)) is \( 2^{2}=4 \) times the area of the original ( \( ABCD \)). So the ratio of \( A'B'C'D' \) to \( ABCD \) is \( 4:1 \). Wait, but let's check again. Let the side length of square \( ABCD \) be \( s \). Area of \( ABCD \) is \( s^{2} \). After dilation by factor 2, side length of \( A'B'C'D' \) is \( 2s \), area is \( (2s)^{2}=4s^{2} \). So the ratio of area of \( A'B'C'D' \) to \( ABCD \) is \( 4s^{2}:s^{2}=4:1 \).

Answer:

C. 4:1