Consider the following statements.
P. If the angles subtended by the two chords at the centre of a circle are equal, then the chords are equal.
Q. If two circles intersect at two points, then the line through the centres is the perpendicular bisector of common chord.
Which of the following options is correct?

Options

A. Only P is true
B. Only Q is true
C. Both P and Q are true
D. Neither P nor Q is true

Answer

To solve the question regarding the statements P and Q, we will analyze each statement step by step.

### Step 1: Analyze Statement P
**Statement P:** If the angles subtended by two chords at the center of a circle are equal, then the chords are equal.

1. **Draw a Circle:** Start by drawing a circle with center O.
2. **Draw Two Chords:** Label the endpoints of the first chord as A and B, and the endpoints of the second chord as C and D.
3. **Draw Angles:** Draw lines OA, OB, OC, and OD to the center O from the endpoints of the chords. This creates angles ∠AOB and ∠COD at the center.
4. **Assume Angles are Equal:** According to the statement, assume that ∠AOB = ∠COD.
5. **Congruent Triangles:** We can show that triangles OAB and OCD are congruent by the Angle-Side-Angle (ASA) criterion because:
- OA = OC (radii of the same circle)
- OB = OD (radii of the same circle)
- ∠AOB = ∠COD (given)
6. **Conclusion:** Since the triangles are congruent, it follows that AB = CD. Therefore, the statement P is true.

### Step 2: Analyze Statement Q
**Statement Q:** If two circles intersect at two points, then the line through the centers is the perpendicular bisector of the common chord.

1. **Draw Two Intersecting Circles:** Draw two circles that intersect at points A and B. Label the centers of the circles as O and O'.
2. **Draw the Common Chord:** Connect points A and B to form the common chord AB.
3. **Draw the Line through the Centers:** Draw a line connecting the centers O and O'.
4. **Perpendicular Bisector:** To prove that the line OO' is the perpendicular bisector of AB, we need to show that it bisects AB at a right angle.
5. **Triangles and Congruence:** Consider triangles OAP and O'BP, where P is the midpoint of AB. By the properties of circles:
- OA = O'A (radii of the circles)
- OB = O'B (radii of the circles)
- AP = PB (P is the midpoint)
6. **Conclusion:** By the Side-Side-Side (SSS) congruence criterion, triangles OAP and O'BP are congruent, which implies that OO' is perpendicular to AB and bisects it. Therefore, statement Q is also true.

### Final Conclusion
Both statements P and Q are true. Therefore, the correct option is that both statements are true.