The radius of the circle is a line from the center of the circle to its circumference
The radius of the circle with the same vertex as a center is 10.16 units
From the question, we understand that:
Start by calculating the area of the triangle using:
[tex]Area = 0.5ab\sin C[/tex]
So, we have:
[tex]Area = 0.5 * 12 * 12\sqrt 3 *\sin (60)[/tex]
This gives
[tex]Area = 0.5 * 12 * 12\sqrt 3 * \frac{\sqrt 3}{2}[/tex]
[tex]Area = 108[/tex]
The arc divides the triangle into two equal regions.
So, the area of the arc is:
[tex]Arc = 0.5 * Area = 0.5 * 108[/tex]
[tex]Arc = 54[/tex]
Also, the area of the arc is:
[tex]Arc = \frac{\theta}{360} * \pi r^2[/tex]
So, we have:
[tex]Arc = \frac{60}{360} *3.14 * r^2[/tex]
Simplify
[tex]Arc = 0.523 * r^2[/tex]
This gives
[tex]0.523 * r^2 = 54[/tex]
Divide both sides by 0.523
[tex]r^2 = 103.25[/tex]
Take the square roots of both sides
[tex]r = 10.16[/tex]
Hence, the radius of the circle with the same vertex as a center is 10.16 units
Read more about circumcised triangles at:
https://brainly.com/question/4268382
