Question

in the figure, △abc is congruent to △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 linear scale factor of dilation is \(k\), the ratio of the areas of the dilated shape \(A_{2}\) to the original shape \(A_{1}\) is \(k^{2}\).

Step2: Identify the scale factor

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

Step3: Calculate the ratio of areas

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

Answer:

C. \(4:1\)