A circular plot of land is divided into two regions by a chord of length 103 meters such that the chord subtends an angle of 120∘ at the center. Then, the area, in square meters, of the smaller region is
Official Correct Answer: C. ### Core Concept
The problem involves finding the area of a smaller segment of a circle divided by a chord. The key concepts are the relationship between the chord length, radius, and central angle, as well as the formulas for the area of a sector and the area of a triangle.
### Step-by-Step Solution
1. **Find the Radius (r):**
The chord length formula is:
\[
\text{Chord length} = 2r \sin\left(\frac{\theta}{2}\right)
\]
Given the chord length is \(10\sqrt{3}\) meters and the central angle \(\theta = 120^\circ\):
\[
10\sqrt{3} = 2r \sin(60^\circ)
\]
Since \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\):
\[
10\sqrt{3} = 2r \cdot \frac{\sqrt{3}}{2} \implies 10\sqrt{3} = r\sqrt{3} \implies r = 10 \text{ meters}
\]
2. **Calculate the Area of the Sector:**
The area of a sector with central angle
Answer and solution
Answer:C) 25(34π−3)
📌Core Concept
The problem involves finding the area of a smaller segment of a circle divided by a chord. The key concepts are the relationship between the chord length, radius, and central angle, as well as the formulas for the area of a sector and the area of a triangle.
