To solve this problem, let's break down each part with details for clarity.

Given Data:

• We have a circle of radius $r = 10$.
• We need to find the area $A$ of the largest isosceles triangle that can be inscribed in this circle.

Approach:

Part (a): Area as a Function of $h$

Let $h$ represent the height of the triangle from the base to the apex (top vertex of the isosceles triangle).

1. Since the triangle is inscribed in a circle with radius $r = 10$, the apex of the triangle will be at the topmost point of the circle.
2. The base of the triangle will be perpendicular to $h$ and will form a chord of the circle.
3. To find the area $A$ as a function of $h$, we can use the fact that: $A = \frac{1}{2} \times \text{base} \times \text{height}$
4. Using trigonometry and the properties of a circle, the base can be expressed in terms of $h$. However, this part requires careful geometry to calculate correctly.

Part (b): Area as a Function of $\alpha$

Let $\alpha$ represent the angle at the apex (the angle subtended by the base at the circle's center).

1. For the largest isosceles triangle, we need to maximize the area in terms of $\alpha$.
2. We know that $A$ can be found using the formula: $A = \frac{1}{2} r^2 \sin(\alpha)$ where $r = 10$.
3. Substitute $r = 10$: $A = \frac{1}{2} \times 10^2 \times \sin(\alpha) = 50 \sin(\alpha)$
4. To maximize $A$, we should find the value of $\alpha$ that gives the maximum $\sin(\alpha)$, which is $\alpha = 90^\circ$ (or $\pi/2$ radians).

Thus, the maximum area $A$ would be: $A = 50$ if the triangle were a right triangle.

Part (c): Type of Triangle of Maximum Area

The type of triangle that maximizes the area, given the conditions, is a right triangle.

So the correct answer should be:

• Part (a): $A = 50$
• Part (b): $A = 50 \sin(\alpha)$ (maximized at $50$ when $\alpha = 90^\circ$)
• Part (c): Right triangle

Would you like additional clarification on any of these parts?

Follow-Up Questions:

1. How can the area be calculated if $h$ is known but not the angle?
2. What is the significance of the radius in determining the maximum area?
3. Can you explain why a right triangle provides the maximum area?
4. How would this problem change if the circle had a different radius?
5. What if the triangle were scalene instead of isosceles—how would that affect the solution?

Tip:

For maximum area problems, it's often helpful to express area formulas in terms of trigonometric functions to take advantage of maximum values at known angles (like $90^\circ$).