If a flag staff subtends the same angles at the points A,B,C and D on the horizontal plane through its foot, the points A,B,C and D from a

Options

A. Square
B. Rectangular
C. Cyclic quadrilaterla
D. None of these

Answer

To solve the problem, we need to analyze the situation where a flagstaff subtends the same angle at four points A, B, C, and D on the horizontal plane. We will determine the geometric configuration of these points.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let the flagstaff be represented by the vertical line segment OP, where O is the foot of the flagstaff on the ground and P is the top of the flagstaff.
- The angles subtended by the flagstaff at points A, B, C, and D on the horizontal plane are all equal to a certain angle θ.

2. **Identifying the Angles**:
- From point A, the angle subtended by the flagstaff is θ.
- Similarly, from points B, C, and D, the angle subtended is also θ.

3. **Using Trigonometry**:
- For point A, we can use the tangent function:
\[ \tan(\theta) = \frac{h}{r_1} \]
where \(h\) is the height of the flagstaff and \(r_1\) is the distance from point A to the foot of the flagstaff O.
- Thus, we can express the height as:
\[ h = r_1 \tan(\theta) \]

4. **Repeating for Other Points**:
- Similarly, for point B:
\[ \tan(\theta) = \frac{h}{r_2} \implies h = r_2 \tan(\theta) \]
- For point C:
\[ \tan(\theta) = \frac{h}{r_3} \implies h = r_3 \tan(\theta) \]
- For point D:
\[ \tan(\theta) = \frac{h}{r_4} \implies h = r_4 \tan(\theta) \]

5. **Equating Heights**:
- Since the height \(h\) is the same for all points, we can equate the expressions:
\[ r_1 \tan(\theta) = r_2 \tan(\theta) = r_3 \tan(\theta) = r_4 \tan(\theta) \]
- Since \(\tan(\theta)\) is the same for all points, we can simplify this to:
\[ r_1 = r_2 = r_3 = r_4 \]

6. **Conclusion**:
- Since all distances \(r_1, r_2, r_3, r_4\) from the foot of the flagstaff O to the points A, B, C, and D are equal, it implies that points A, B, C, and D lie on a circle with center O.
- Therefore, the points A, B, C, and D form a cyclic quadrilateral.

### Final Answer:
The points A, B, C, and D form a cyclic quadrilateral.

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