In the given figure, $ABC$ is a right-angled triangle with $B$ as the right angle. Three semi-circles are drawn with $AB$, $BC$ and $AC$ as diameters. What is the area of the shaded portion if the area of the triangle $ABC$ is $12$ square units?

### Concept & Lunes of Alhazen The shaded regions are known as the Lunes of Alhazen. The geometric theorem states that if semi-circles are drawn on the three sides of a right-angled triangle, the sum of the areas of the two lunes (crescent shapes) formed on the legs is exactly equal to the area of the right-angled triangle itself. Formula: $$ \text{Area of Lunes} = \text{Area of Right Triangle} $$ ### Step-by-Step Solution 1. Understand the geometry: The total combined area consists of the triangle and the two semi-circles on the legs $AB$ and $BC$. 2. The shaded region is formed by subtracting the unshaded portion of the largest semi-circle (on hypotenuse $AC$) from the total area. 3. Mathematically, Area(shaded) = Area(semi-circle on $AB$) + Area(semi-circle on $BC$) + Area($\triangle ABC$) - Area(semi-circle on $AC$). 4. By Pythagoras theorem, $AB^2 + BC^2 = AC^2$. Therefore, the sum of areas of the semi-circles on $AB$ and $BC$ equals the area of the semi-circle on $AC$. 5. Canceling these out leaves: Area(shaded) = Area($\triangle ABC$). 6. Since Area($\triangle ABC$) = $12$ square units, Area(shaded) = $12$ square units. ### Exam Strategy & Shortcut For any right-angled triangle with semi-circles drawn outwards on its sides and the hypotenuse semi-circle folded back over the triangle, the area of the resulting crescent moons always equals the area of the triangle. Memorize this fact to answer instantly without any calculation. ### Common Pitfall Attempting to calculate the lengths of the sides and individual areas is impossible because only the total triangle area is given. Not knowing the specific theorem leads to incorrectly choosing "Cannot be determined". ### Final Answer Therefore, the correct answer is **12 square units**.