Q1. Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.
Solution:
Given: Two congruent circles with centers \( O \) and \( O' \), each having radius \( r \). Two chords \( PQ \) and \( P'Q' \) such that \( PQ = P'Q' \).
To Prove: The angles subtended by chords \( PQ \) and \( P'Q' \) at the centers of their respective circles are equal, i.e., \( \angle POQ = \angle P'O'Q' \).
Proof:
Consider triangles \( \triangle OPQ \) and \( \triangle O'P'Q' \).
- Since both circles are congruent, the radii are equal:
\( OP = O'P' = OR = O'Q' = r \) - The chords are equal by given condition:
\( PQ = P'Q' \) - The side \( OP = O'P' \) (radius) and \( OQ = O'Q' \) (radius) are equal.
Therefore, by the SSS (Side-Side-Side) congruence criterion, we have:
\[
\triangle OPQ \cong \triangle O'P'Q'
\]
Consequently, corresponding angles of congruent triangles are equal by CPCT (Corresponding Parts of Congruent Triangles): \[ \therefore \angle POQ = \angle P'O'Q' \]
Hence, equal chords of congruent circles subtend equal angles at their centers.
Q2. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.
Solution:
Given: Two congruent circles with centers \( A \) and \( P \), both having equal radius. Two chords \( BC \) and \( QR \) subtending equal angles at their centers, i.e.,
\[ \angle BAC = \angle QPR \]
To Prove: The chords \( BC \) and \( QR \) are equal, i.e.,
\[ BC = QR \]
Proof:
Consider triangles \( \triangle ABC \) and \( \triangle PQR \), where \( A \) and \( P \) are the
centers of
congruent circles.
- Radii of congruent circles are equal: \[ AB = PQ \quad \text{and} \quad AC = PR \]
- Given angles subtended at the centers are equal: \[ \angle BAC = \angle QPR \]
By the SAS (Side-Angle-Side) congruence criterion, we have:
\[ \triangle ABC \cong \triangle PQR \]
Therefore, by CPCT (Corresponding Parts of Congruent Triangles):
\[ BC = QR \]
Hence, chords of congruent circles subtending equal angles at their centres are equal.