To find the area of the shaded part of the given geometric figure, we will analyze the shape based on the image provided.

Analysis of the Shape:

1. Triangle $ABC$:

   • The triangle $ABC$ is an isosceles triangle where $AB = AC$, as indicated by the markings.

2. Circles:

   • There are two semicircles inside the triangle.
   • Both semicircles have their diameters on line $BC$.

3. Shaded Area:

   • The shaded area is the part of the triangle that is not covered by the semicircles.

Steps to Calculate the Shaded Area:

1. Calculate the Area of Triangle $ABC$:

   • Since $AB = AC$ and assuming $\angle BAC = 90^\circ$, we can use the following formula for the area of a right-angled triangle: $\text{Area}_{\triangle ABC} = \frac{1}{2} \times AB \times AC$

2. Calculate the Area of the Semicircles:

   • Each semicircle has a radius $r = 16$ cm.
   • The area of one semicircle is $\frac{1}{2} \times \pi r^2$.
   • The total area of two semicircles is: $2 \times \left(\frac{1}{2} \times \pi \times 16^2\right) = \pi \times 16^2$

3. Calculate the Shaded Area:

   • The shaded area is the area of the triangle minus the area of the two semicircles: $\text{Shaded Area} = \text{Area}_{\triangle ABC} - \pi \times 16^2$

Calculations:

1. Area of the Semicircles: $\pi \times 16^2 = 3.14 \times 256 = 803.84 \, \text{cm}^2$

2. Approximation (assuming $AB = AC = BC = 32$ for simplicity):

   • If triangle $ABC$ is a right-angled isosceles triangle, then: $\text{Area}_{\triangle ABC} = \frac{1}{2} \times 32 \times 32 = 512 \, \text{cm}^2$

3. Shaded Area: $\text{Shaded Area} = 512 - 803.84 = -291.84 \, \text{cm}^2$

Conclusion:

Since the shaded area is negative, there seems to be an inconsistency in the assumption about the triangle’s dimensions. It is important to verify the values given and the exact orientation and calculation of the triangle. Typically, if $AB = AC$, then the base $BC$ might not be equal to $32$, or $\angle BAC$ isn't $90^\circ$, and further measurements or assumptions are needed. Please confirm any additional information or check the triangle dimensions if the result seems incorrect.

Let me know if you want further assistance or adjustments based on different assumptions or data!

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Additional Related Questions

1. How do we find the area of a circle?
2. What are the properties of an isosceles triangle?
3. How does the Pythagorean theorem apply to right triangles?
4. What is the formula for the circumference of a circle?
5. Can the area of complex shapes be found using integration?
6. How can symmetry help in solving geometric problems?
7. What is the significance of the centroid in triangles?
8. How are semicircles different from full circles in terms of area and perimeter?

Tip:

Always ensure to check the assumptions made in geometric problems. Accurate measurements or clear descriptions can greatly affect the result.