Question

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 area - dilation formula

For a two - dimensional shape, if the scale factor of dilation is \(k\), the ratio of the area of the dilated shape \(A_{2}\) to the area of the original shape \(A_{1}\) is \(k^{2}\).

Step2: Identify the scale factor

The square \(ABCD\) is dilated by a factor of \(k = 2\) to form \(A'B'C'D'\).

Step3: Calculate the ratio of the areas

Using the formula \(\frac{A_{A'B'C'D'}}{A_{ABCD}}=k^{2}\), substituting \(k = 2\), we get \(k^{2}=2^{2}=4\). So the ratio of the area of \(A'B'C'D'\) to the area of \(ABCD\) is \(4:1\).

Answer:

C. \(4:1\)