Introduction:
In this problem, we are given an isosceles triangle inscribed in a circle of radius 10 cm. We need to find the value of the vertex angle that maximizes the area of the triangle.

Understanding the Problem:
To find the value of the vertex angle that maximizes the area of the triangle, we first need to understand the properties of an isosceles triangle inscribed in a circle.

Properties of Isosceles Triangle Inscribed in a Circle:
1. The base angles of an isosceles triangle (the angles opposite to the equal sides) are congruent.
2. The vertex angle (the angle formed by the two equal sides) is twice the measure of each base angle.
3. The base of the isosceles triangle is also the diameter of the circle.

Approach:
To maximize the area of the triangle, we need to maximize the length of the base, which is also the diameter of the circle. Let's assume the vertex angle of the triangle is θ.

Calculating the Length of the Base:
Since the base of the triangle is the diameter of the circle, we can use the formula for the circumference of a circle to find the length of the base.

Circumference = 2πr

In this case, the radius (r) is given as 10 cm. So, the length of the base is:

Length of Base = 2π(10) = 20π cm

Calculating the Measure of the Base Angle:
Since the base angles of an isosceles triangle are congruent, we can calculate the measure of each base angle using the formula for the sum of angles in a triangle.

Sum of Angles in a Triangle = 180°

In this case, the vertex angle is 2θ. So, the measure of each base angle is:

(180° - 2θ) / 2 = 90° - θ

Calculating the Area of the Triangle:
Now, we can calculate the area of the isosceles triangle using the formula:

Area = (1/2) * base * height

In this case, the height of the triangle is equal to the radius of the circle, which is 10 cm. So, the area of the triangle is:

Area = (1/2) * 20π * 10 = 100π cm²

Maximizing the Area:
To maximize the area of the triangle, we need to find the value of θ that maximizes the area. We can do this by taking the derivative of the area formula with respect to θ and setting it to zero.

Taking the derivative:

d(Area)/dθ = 0

Simplifying the equation and solving for θ will give us the value that maximizes the area.

Conclusion:
By applying the properties of an isosceles triangle inscribed in a circle, we can determine that the value of the vertex angle that maximizes the area of the triangle is obtained when the base angle is 45°. At this angle, the length of the base is 20π cm and the area of the triangle is 100π cm².