Question

A triangle has a 60 degree angle, and the two adjacent sides are 12 and 12 times the square root of 3. Find the radius of a circle with the same vertex as a center, if the arc inside the triangle divides it into two regions of equal area.
Please somebody ?

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Answer 1

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

How to determine the radius

From the question, we understand that:

• The adjacent sides of the triangle are 12 and $12\sqrt 3$
• The triangle has an angle of 60 degrees

Start by calculating the area of the triangle using:

$Area = 0.5ab\sin C$

So, we have:

$Area = 0.5 * 12 * 12\sqrt 3 *\sin (60)$

This gives

$Area = 0.5 * 12 * 12\sqrt 3 * \frac{\sqrt 3}{2}$

$Area = 108$

The arc divides the triangle into two equal regions.

So, the area of the arc is:

$Arc = 0.5 * Area = 0.5 * 108$

$Arc = 54$

Also, the area of the arc is:

$Arc = \frac{\theta}{360} * \pi r^2$

So, we have:

$Arc = \frac{60}{360} *3.14 * r^2$

Simplify

$Arc = 0.523 * r^2$

This gives

$0.523 * r^2 = 54$

Divide both sides by 0.523

$r^2 = 103.25$

Take the square roots of  both sides

$r = 10.16$

Hence, the radius of the circle with the same vertex as a center is 10.16 units

Read more about circumcised triangles at:

brainly.com/question/4268382