The problem describes a triangle $\triangle ABC$ with sides $AC = 3$, $BC = 5$, and $AB = 7$. Each side is the diameter of a semicircle, and the shaded areas $S_1$, $S_2$, and $S_3$ correspond to these semicircles. The question asks for $S_1 + S_3 - S_2$, where $S_1$, $S_2$, and $S_3$ are the areas of the grey semicircles as shown in the diagram.

Step 1: Area of a semicircle

The area $A$ of a semicircle is given by: $A = \frac{1}{2} \pi r^2$ where $r$ is the radius of the circle, which is half the diameter.

Step 2: Calculate the areas $S_1$, $S_2$, and $S_3$

The diameters of the semicircles are the sides of the triangle.

• For $AC = 3$:

  • Radius = $\frac{3}{2}$
  • Area $S_1 = \frac{1}{2} \pi \left(\frac{3}{2}\right)^2 = \frac{1}{2} \pi \times \frac{9}{4} = \frac{9\pi}{8}$

• For $BC = 5$:

  • Radius = $\frac{5}{2}$
  • Area $S_2 = \frac{1}{2} \pi \left(\frac{5}{2}\right)^2 = \frac{1}{2} \pi \times \frac{25}{4} = \frac{25\pi}{8}$

• For $AB = 7$:

  • Radius = $\frac{7}{2}$
  • Area $S_3 = \frac{1}{2} \pi \left(\frac{7}{2}\right)^2 = \frac{1}{2} \pi \times \frac{49}{4} = \frac{49\pi}{8}$

Step 3: Compute $S_1 + S_3 - S_2$

Now, we calculate: $S_1 + S_3 - S_2 = \frac{9\pi}{8} + \frac{49\pi}{8} - \frac{25\pi}{8}$ $S_1 + S_3 - S_2 = \frac{9\pi + 49\pi - 25\pi}{8} = \frac{33\pi}{8}$

Step 4: Compare with the options

The result $\frac{33\pi}{8}$ is not directly one of the provided options. However, this might be a simplified or approximated form of one of the answers. After checking, it doesn't match exactly with the options, so there might be an alternative approach involving simplifying expressions with square roots from the triangle properties.

Let me know if you'd like further clarifications or if you'd like to explore other methods!

Do you want more details on solving this? Here are 5 related questions:

1. How is the area of a full circle derived from the formula for a semicircle?
2. Why is $S_2$ larger than both $S_1$ and $S_3$?
3. How can the Pythagorean theorem apply to this triangle setup?
4. What geometric properties of semicircles are used in this problem?
5. Can this approach be extended to other polygons with similar configurations?

Tip: When dealing with semicircles, always halve both the radius and the area formula from a full circle to avoid calculation mistakes.