Exercise 9.3: Chord and Arcs of a Circle

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Chapter Overview & SLOs

How do geometric links between matching boundary arcs and internal straight-line chords establish rigorous proof systems for cyclic quadrilaterals and inscribed triangles? The final section of Chapter 9 explores the operational dualities of circles, where structural curvatures (arcs) translate directly into linear segments (chords) and corresponding angles.

The Arc-Chord Congruence Theorem: Within a single circle (or across matching congruent circles):

• Equal chords subtend equal arcs: $AB = CD$ → $\hat{AB} = \hat{CD}$
• Equal arcs subtend equal chords: $\hat{AB} = \hat{CD}$ → $AB = CD$
• If arc $\hat{AB} < \hat{AD}$, then chord $AB <$ chord $AD$

Symmetric Chord Partitions in Cyclic Quadrilaterals: For cyclic quadrilateral $ABDC$ with $AB = CD$:

• Step 1: $AB = CD$ → $\hat{AB} = \hat{CD}$ (equal chords → equal arcs)
• Step 2: Add shared arc $\hat{AD}$ to both sides: $\hat{AB} + \hat{AD} = \hat{CD} + \hat{AD}$
• Step 3: This gives $\overline{BAD} = \overline{CDA}$ (larger overlapping arcs)
• Step 4: Equal arcs → equal chords: $BD = AC$
• Therefore, diagonals $AC$ and $BD$ are equal.

Inscribed Isosceles Triangle Angle Bisector Proof: For isosceles $\triangle ABC$ with $AB = AC$ inscribed in a circle:

• $AB = AC$ → $\hat{AB} = \hat{AC}$ (equal chords → equal arcs)
• Angles subtended by equal arcs are equal: $\angle BDA = \angle CDA$
• Therefore, $AD$ bisects $\angle BDC$

Key Theorems for Proofs:

• Equal chords subtend equal arcs
• Equal arcs subtend equal chords
• Angles in the same segment are equal
• Equal arcs subtend equal angles at the circumference

Important Exam Note: Always state the supporting theorem in parentheses next to each geometric statement in proofs.

These notes are strictly aligned with the Student Learning Outcomes (SLOs) for the FBISE 2026 annual examination.

• How do we approve and apply the fundamental geometric axiom that equal chords always subtend equal boundary arcs within circular systems?
• How do we execute geometric proofs for cyclic quadrilaterals by applying additive properties to partitioned arc systems?
• How do we prove angle bisector parameters ($\angle BDA = \angle CDA$) for inscribed triangles using matching perimeter arc sectors?
• How do we deduce chord inequality constraints from given arc curvature relationships to evaluate non-symmetric circular fields?

Frequently Asked Questions (FAQ)