Question

circle d and radius \\( \overline { c d } \\) are shown on the graph below. circle d is dilated by a scale factor of 0.4 centered at point d to create circle \\( d ^ { prime } \\).

what is the area of circle \\( d ^ { prime } \\)?

write your answer as a whole number or as a decimal rounded to the nearest tenth.

\\( \square \\) square units

Explanation:

Step1: Find the length of radius \( CD \)

Using the distance formula (or counting units on the graph), if \( C(-8,6) \) and \( D(-6,3) \), then the length of \( CD=\sqrt{(-8 + 6)^{2}+(6 - 3)^{2}}=\sqrt{(-2)^{2}+3^{2}}=\sqrt{4 + 9}=\sqrt{13}\approx3.6 \) units.

Step2: Find the radius of the dilated circle \( D' \)

Since the scale factor is \( 0.4 \), the radius of \( D' \), \( r'=0.4\times r \). Substituting \( r = \sqrt{13}\), we get \( r'=0.4\times3.6 = 1.44 \) units.

Step3: Calculate the area of circle \( D' \)

The formula for the area of a circle is \( A=\pi r^{2} \). Substituting \( r' = 1.44 \), we have \( A=\pi\times(1.44)^{2}\approx3.14\times2.0736\approx6.5 \).

Answer:

\( 6.5 \)