$ABC$ is a right-angled triangle with right angle at $B$. If the semi-circle on $AB$ with $AB$ as diameter encloses an area of 81 sq. cm and the semi-circle on $BC$ with $BC$ as diameter encloses an area of 36 sq. cm, then the area of the semi-circle on $AC$ with $AC$ as diameter will be

### Concept & Generalized Pythagorean Theorem The Pythagorean theorem states that in a right triangle, $a^2 + b^2 = c^2$. This principle generalizes to any similar geometric figures constructed on the sides of a right triangle. If you draw similar shapes (like squares, equilateral triangles, or semi-circles) on the three sides, the area of the shape on the hypotenuse equals the sum of the areas of the shapes on the two legs. $$\text{Area}_{AC} = \text{Area}_{AB} + \text{Area}_{BC}$$ ### Step-by-Step Solution 1. **Identify the Components:** - $\text{Area of semi-circle on leg } AB = 81$ sq. cm. - $\text{Area of semi-circle on leg } BC = 36$ sq. cm. - $AC$ is the hypotenuse (since the right angle is at $B$). 2. **Apply the Generalized Pythagorean Theorem:** The area of the semi-circle on the hypotenuse ($AC$) is the sum of the areas of the semi-circles on the legs ($AB$ and $BC$). $\text{Area}_{AC} = 81 + 36$ 3. **Calculate the Final Area:** $\text{Area}_{AC} = 117$ sq. cm. ### Exam Strategy & Shortcut Recognize that area scales with the square of the dimension. Since $AB^2 + BC^2 = AC^2$, multiplying the entire equation by $\frac{\pi}{8}$ gives the semi-circle areas: $\frac{\pi}{8}AB^2 + \frac{\pi}{8}BC^2 = \frac{\pi}{8}AC^2$. Thus, you just add the two given areas directly: $81 + 36 = 117$. No need to find radii. ### Common Pitfall A massive time-waster is working backward to find the actual radii of the semi-circles, computing the side lengths of the triangle, using the Pythagorean theorem to find $AC$, and then computing the area of the semi-circle on $AC$. While correct, it takes ten times longer. ### Final Answer Therefore, the correct answer is **117 cm$^2$**.