The problem presented is based on the properties of an inscribed angle in a circle. The theorem mentioned states that the measure of an inscribed angle is half the measure of its intercepted arc, and here we are using that concept to show that angle $\angle C$ is a right angle ($90^\circ$).

Key Details from the Problem:

1. $\overline{AB}$ is the diameter of the circle, and we know that any inscribed angle that intercepts a semicircle (formed by a diameter) must be $90^\circ$.
2. The coordinates for points $A$, $B$, and $C$ are given:
   • $A(-c, 0)$
   • $B(c, 0)$
   • $C(a, b)$
3. We are asked to choose which equation helps to prove that $m\angle C = 90^\circ$.

Step-by-step Solution:

To determine that $m\angle C = 90^\circ$, we can utilize the slope-based method.

• The slopes of lines $\overline{AC}$ and $\overline{BC}$ are involved. If the product of their slopes equals $-1$, the two lines are perpendicular, meaning $\angle C = 90^\circ$.

1. Slope of line $\overline{AC}$ is given by: $\text{slope of } \overline{AC} = \frac{b - 0}{a - (-c)} = \frac{b}{a + c}$

2. Slope of line $\overline{BC}$ is given by: $\text{slope of } \overline{BC} = \frac{b - 0}{a - c} = \frac{b}{a - c}$

3. For $\overline{AC}$ and $\overline{BC}$ to be perpendicular, the product of these slopes should equal $-1$: $\left( \frac{b}{a + c} \right) \times \left( \frac{b}{a - c} \right) = -1$

Conclusion:

The correct equation that helps in proving that $m\angle C = 90^\circ$ is: $\left( \frac{b}{a - c} \right) \times \left( \frac{b}{a + c} \right) = -1$ This matches the second option in the list.

Would you like more detailed explanations or have any further questions? Here are a few related questions to deepen understanding:

1. Why does an inscribed angle subtended by a diameter always measure $90^\circ$?
2. What are the slopes of two perpendicular lines in general geometry?
3. How can the equation involving slopes be derived geometrically from the circle?
4. Can we generalize this method to any inscribed triangle in a circle?
5. How does the Pythagorean theorem relate to the geometry of this circle?

Tip: For problems involving circles, always look for opportunities to apply known circle theorems like the inscribed angle theorem and properties of diameters.