In the adjoining figure, $\triangle OAB$ is an equilateral triangle and the area of the shaded region is $750\pi$ cm$^2$. Find the perimeter of the shaded region.
Show Hint
In area-ratio problems, if the radii and central angles are identical, the areas will always be equal regardless of the orientation of the figure.
Basic Idea:
The shaded portion represents a major sector of a circle. In an equilateral triangle, each central angle is $60^\circ$. Therefore, the remaining part of the circle forms the major sector. The area and perimeter can be calculated using standard sector formulas.
Formulas Used:
1) Area of sector = $\frac{\theta}{360} \times \pi r^2$
2) Arc length = $\frac{\theta}{360} \times 2\pi r$
Complete Calculation:
1) Since the minor angle is $60^\circ$, the major sector angle = $360^\circ - 60^\circ = 300^\circ$.
2) Area of major sector = $\frac{300}{360} \pi r^2 = \frac{5}{6} \pi r^2$.
3) Given area = $750\pi$
So, $\frac{5}{6} r^2 = 750$
$r^2 = 750 \times \frac{6}{5} = 900$
$r = 30$ cm.
4) Arc length of major sector = $\frac{300}{360} \times 2\pi (30)$
$= \frac{5}{6} \times 60\pi$
$= 50\pi$ cm.
5) Perimeter of shaded region = Arc length + two radii
$= 50\pi + 2(30)$
$= 50\pi + 60$ cm.
Final Result:
Radius = 30 cm
Perimeter of shaded region = $(50\pi + 60)$ cm.
