If a tower subtends equal angles at four points P, Q , R and S that lie in a plane containing the foot of the tower , the which fo the following statements is always true (here, the tower is perpendicular to the plane containing the points P, Q,R,S)

Options

A. `anglePQS=anglePRS`
B. `anglePQR+anglePSR=180^@`
C. `anglePQS=90^@rArranglePRS=90^@`
D. `(PQ)(RS)+(PS)(RQ)=(PR)(QS)`

Answer

To solve the problem, we need to analyze the situation where a tower subtends equal angles at four points P, Q, R, and S that lie in a plane containing the foot of the tower. The tower is perpendicular to this plane.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- Let the foot of the tower be point O and the top of the tower be point B. The height of the tower is denoted as \( h \).
- The points P, Q, R, and S are located in the horizontal plane at distances from point O.

2. **Equal Angles Subtended**:
- Since the tower subtends equal angles at points P, Q, R, and S, we denote this angle as \( \theta \). Thus, we have:
\[ \angle POB = \angle QOB = \angle ROB = \angle SOB = \theta \]

3. **Using Trigonometry**:
- From the definition of tangent in a right triangle, we can express the distances OP, OQ, OR, and OS in terms of \( h \) and \( \theta \):
\[ OP = h \tan(\theta), \quad OQ = h \tan(\theta), \quad OR = h \tan(\theta), \quad OS = h \tan(\theta) \]
- This indicates that all points P, Q, R, and S are equidistant from point O.

4. **Cyclic Quadrilateral**:
- Since all four points are equidistant from O, they lie on a circle centered at O. Therefore, quadrilateral PQRS is a cyclic quadrilateral.
- A property of cyclic quadrilaterals is that the sum of the opposite angles is equal to 180 degrees:
\[ \angle PQR + \angle PSR = 180^\circ \quad \text{and} \quad \angle PQS + \angle PRS = 180^\circ \]

5. **Analyzing the Options**:
- **Option 1**: \( \angle PQS = \angle PRS \) - This is not necessarily true.
- **Option 2**: \( \angle PQR + \angle PSR = 180^\circ \) - This is true for cyclic quadrilaterals, but it may not always hold depending on the order of points.
- **Option 3**: \( \angle PQS = 90^\circ \) implies \( \angle PRS = 90^\circ \) - This is true because both angles subtend the same chord PS.
- **Option 4**: The statement about products of sides and diagonals is not always true.

6. **Conclusion**:
- The only statement that is always true is **Option 3**: \( \angle PQS = 90^\circ \) implies \( \angle PRS = 90^\circ \).

### Final Answer:
The correct statement that is always true is:
**Option 3: \( \angle PQS = 90^\circ \) implies \( \angle PRS = 90^\circ \)**.