Question

In the given figure, triangle $ABC$ is a right-angled triangle in which \[\angle A = {90^ \circ }\], semi circles are drawn on $AB, AC, BC$ as diameters. find the area of the shaded region

Answer 1

Hint:As the question mentions that this is a right-angled triangle, so we can apply the Pythagoras theorem, and calculate the length of that unknown side, then by using the formula, the area of the entire triangle ABC can be calculated further subtracted arbitrarily as shown below.

Formula used:
Area of a triangle = \[\dfrac{1}{2} \times base \times height\] and Area of a semicircle= \[\dfrac{1}{2} \times \pi {r^2}\].

Complete step by step answer:
In the given figure \[\vartriangle ABC\], \[\angle A = {90^ \circ }\]. By applying the Pythagoras theorem, we calculate the side BC. The theorem says that in a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides. The side opposite to the right angle (90°) is the longest side (known as Hypotenuse).
\[B{C^2} = A{B^2} + A{C^2} \\
\Rightarrow B{C^2} = {3^2} + {4^2} \\
\Rightarrow 9 + 16 = 25 \\ \]
Therefore, \[BC = 5\,cm\]

We can calculate the area of the shaded region as:
\[\text{Area of} \Delta ABC +\text{area of semicircle AB + area of semicircle AC - area of semicircle BAC}\]
Putting the values as such:
\[\text{Area of shaded region}= \dfrac{1}{2} \times 3 \times 4 + \dfrac{1}{2} \times \pi {\left( {\dfrac{3}{2}} \right)^2} + \dfrac{1}{2} \times \pi {\left( {\dfrac{4}{2}} \right)^2} - \dfrac{1}{2} \times \pi {\left( {\dfrac{5}{2}} \right)^2}\]
\[\Rightarrow \text{Area of shaded region}= 6 + \dfrac{{9\pi }}{8} + 2\pi - \,\dfrac{{25\pi }}{8} \\
\Rightarrow \text{Area of shaded region}= 6 + 2\pi - 2\pi \\
\therefore \text{Area of shaded region}= 6\,c{m^2} \\ \]
Therefore, the area of the shaded region is \[6\,c{m^2}\].

Note: This was a question based on areas and perimeters. Here we had to calculate the area of a chunk of a figure so, we calculated the area of the whole figure and the area that excluded that chunk of the figure (whose area is to be calculated), subtracted them.