Question

point g is the center of the small circle. point x is the center of the large circle. points g, y, and x are all on line segment gx. marco wants to create a new circle using \\( \overline { g x } \\) as a radius. what will be the area of marcos new circle? \\( 10 \pi \mathrm { cm } ^ { 2 } \\) \\( 16 \pi \mathrm { cm } ^ { 2 } \\) \\( 356 \pi \mathrm { cm } ^ { 2 } \\) \\( 676 \pi \mathrm { cm } ^ { 2 } \\)

Explanation:

Step1: Calculate the length of \( GX \)

Since \( GY = 10\mathrm{cm} \) and \( YX=16\mathrm{cm} \), then \( GX=GY + YX\).
\( GX=10 + 16=26\mathrm{cm} \)

Step2: Use the formula for the area of a circle

The formula for the area of a circle is \( A=\pi r^{2} \), where \( r \) is the radius. Here \( r = GX = 26\mathrm{cm} \).
Substitute \( r = 26 \) into the formula: \( A=\pi\times(26)^{2}\).
\( A=\pi\times676 = 676\pi\mathrm{cm}^{2}\)

Answer:

\(676\pi\mathrm{cm}^{2}\) (the fourth option)