Question

in the diagram, the circle will be dilated by a scale factor of 3 about the origin. the points c, a, and b map to c, a, and b after the dilation. what is the length of \\(\overline{cb}\\)? use the distance formula to help you decide: \\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\).
a. 45 units
b. 24 units
c. 21 units
d. 5 units
e. 15 units

Explanation:

Step1: Find length of \( CB \)

Use distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) for \( C(8,10) \) and \( B(12,13) \).
\( d = \sqrt{(12 - 8)^2 + (13 - 10)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \) units.

Step2: Apply dilation scale factor

Dilation by scale factor \( 3 \) about origin. Lengths scale by factor \( 3 \).
Length of \( C'B' = 3 \times \) length of \( CB = 3 \times 5 = 15 \)? Wait, no—wait, wait, no, wait: Wait, no, wait, the options have 15? Wait, no, wait, maybe I miscalculated \( CB \)? Wait, \( C(8,10) \), \( B(12,13) \): \( x \)-difference \( 12 - 8 = 4 \), \( y \)-difference \( 13 - 10 = 3 \), so \( \sqrt{4^2 + 3^2} = 5 \). Then dilation factor 3: \( 5 \times 3 = 15 \)? But option E is 15. Wait, but let me check again. Wait, maybe the original \( CB \) is 5, so after dilation, 15. So the answer should be E? Wait, no, wait, the options: A.45, B.24, C.21, D.5, E.15. So yes, \( 5 \times 3 = 15 \).

Answer:

E. 15 units