MathematicsChapter 9: Chord and Arcs of a Circle
Chord Theorems At A Glance
Keep the names and the one-line conclusion.
Learning objectives
- Explain chord theorems at a glance using the correct definitions, conditions, and notation.
- Apply bisect <-> perp, equal <-> equidistant correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
What this grid covers
Keep the names and the one-line conclusion. Proofs live in the textbook; the exam often wants the statement plus a Pythagoras number.
Where / symbols
- CHORD THEOREMS AT A GLANCE
- Keep the names and the one-line conclusion. Proofs live in the textbook; the exam often wants the statement plus a Pythagoras number.
- Unique circle
- If 3 non-collinear points, then exactly one circumcircle.
- Bisector from O
- If Line from O bisects chord, then it is perpendicular to the chord.
- Perp from O
- If Perp from O to chord, then it bisects the chord.
- Equal chords
- If AB = CD, then equal distances from O.
WHAT — CHORD THEOREMS AT A GLANCE
Keep the names and the one-line conclusion. Proofs live in the textbook; the exam often wants the statement plus a Pythagoras number. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Chord Theorems At A Glance matters because chord and arc theorems convert circle diagrams into equal lengths, equal angles, and right triangles. They are the foundation for later tangent proofs and constructions.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through read the given condition, select the matching theorem, then apply its conclusion. Worked: a chord is 24 cm, r = 13 cm. Can d be found? m = 12 r^2 = d^2 + m^2 169 = d^2 + 144 d^2 = 25 d = 5 cm Finish by checking signs, restrictions, units, and whether the answer matches the question.
School list:
| Theorem idea | If you know | Then you get |
|---|---|---|
| 1. Unique circle | 3 non-collinear points | Exactly one circumcircle |
| 2. Bisector from O | Line from O bisects chord | It is perpendicular to the chord |
| 3. Perp from O | Perp from O to chord | It bisects the chord |
| 4. Equal chords | AB = CD | Equal distances from O |
| 5. Equal distances | Same d from O | Chords equal |
| 6. Equal arcs/angles | Equal arcs or equal angle at O | Equal chords (matching case) |
Worked example
- a chord is 24 cm, r = 13 cm. Can d be found?
- m = 12
- r2 = d2 + m2
- 169 = d2 + 144
- d2 = 25
- d = 5 cm
In simple words
Quote the matching converse. A theorem and its converse are two exam sentences, not one.
Key points
- bisect <-> perp
- equal <-> equidistant
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Chord Theorems At A Glance
Sticky Notes · Quick Revision
Core Idea
- Keep the names and the one-line conclusion. Proofs live in the textbook; the exam often wants the statement plus a Pythagoras number.
Key Rule
- Unique circle: 3 non-collinear points -> Exactly one circumcircle
- Bisector from O: Line from O bisects chord -> It is perpendicular to the chord
- Perp from O: Perp from O to chord -> It bisects the chord
Second Step
- Equal chords: AB = CD -> Equal distances from O
- Equal distances: Same d from O -> Chords equal
- Equal arcs/angles: Equal arcs or equal angle at O -> Equal chords (matching case)
Worked Method
- Worked: a chord is 24 cm, r = 13 cm. Can d be found?
- m = 12
- r2 = d2 + m2
- 169 = d2 + 144
- d2 = 25
- d = 5 cm
Common Misconception
- Quote the matching converse. A theorem and its converse are two exam sentences, not one.
FBISE Exam Check
- Quote the matching converse. A theorem and its converse are two exam sentences, not one.
60-second revision
- Keep the names and the one-line conclusion.
- Unique circle: 3 non-collinear points gives exactly one circumcircle.
- Bisector from O: Line from O bisects chord gives it is perpendicular to the chord.
- Perp from O: Perp from O to chord gives it bisects the chord.
- Worked-method reminder: d = 5 cm
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