Subject: General Aptitude · Chapter: Area · Exam: · Marks: · Difficulty:
In the given figure, ABC is a B right-angled triangle with B as the right angle. Three semi-circles are drawn with AB, BC and AC as C A diameters. What is the area of the shaded portion if the area of the triangle ABC is 12 square units?
(a)6 square units
(b)12 square units
(c)24 square units
(d)C a n n o t b e d e t e r m i n e d a s t h e d a t a i s insufficient
Answer
Answer (as printed): B
Explanation
Area of the shaded portion = Area of semi-circle with $A B$ as diameter + Area of semi-circle with $B C$ as diameter + Area $(\triangle A B C)$ - Area of semi-circle with $A C$ as diameter $$\begin{aligned} & =\frac{1}{2} \pi\left(\frac{A B}{2}\right)^{2}+\frac{1}{2} \pi\left(\frac{B C}{2}\right)^{2}+12-\frac{1}{2} \pi\left(\frac{A C}{2}\right)^{2} \\ = & \frac{1}{8} \pi\left[(A B)^{2}+(B C)^{2}-(A C)^{2}\right]+12=\frac{1}{8} \pi\left[(A C)^{2}-(A C)^{2}\right]+12 \\ = & 12 \text { sq. units. } \quad\left[\because(A B)^{2}+(B C)^{2}=(A C)^{2}\right] \end{aligned}$$
Explanation as extracted from the printed page; notation may be imperfect.