If the circumcentre of PQR triangle is 'O' and `OX_|_PQ`. What the relation between PX and QX is ?

Options

A. `PX gt QX`
B. `PX lt QX`
C. `PX =QX`
D. `PX=(QX)/(2)`

Answer

To solve the problem, we need to analyze the relationship between the segments PX and QX when the circumcenter O of triangle PQR is such that OX is perpendicular to line segment PQ.

### Step-by-step Solution:

1. **Understanding the Setup**:
- We have a triangle PQR inscribed in a circle, with O as the circumcenter.
- OX is a line segment from O to PQ that is perpendicular to PQ.

2. **Drawing the Diagram**:
- Draw a circle and place triangle PQR inside it.
- Mark the circumcenter O.
- Draw line segment OX such that it meets PQ at point X, and OX is perpendicular to PQ.

3. **Applying Circle Properties**:
- One important property of circles is that if a line from the center of the circle is perpendicular to a chord, it bisects that chord.
- Since OX is perpendicular to PQ, it means that point X divides the chord PQ into two equal segments.

4. **Defining the Segments**:
- Let PX be the segment from point P to point X.
- Let QX be the segment from point Q to point X.

5. **Establishing the Relationship**:
- Since OX bisects PQ, we have PX = QX.

6. **Conclusion**:
- Therefore, the relationship between PX and QX is that they are equal.

### Final Answer:
PX = QX