Chord Theorems at a Glance

Chord Theorems at a Glance

What is it?

Keep the names and the one-line conclusion. Proofs live in the textbook; the exam often wants the statement plus a Pythagoras number.

In simple words

• 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.

Important terms

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 you should learn

• 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.

Full school-level explanation

Keep the names and the one-line conclusion. Proofs live in the textbook; the exam often wants the statement plus a Pythagoras number.

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.

Remember

• bisect <-> perp
• equal <-> equidistant
• d = 5 cm
• converse is separate
• bisect <-> perp
• equal <-> equidistant
• bisect <-> perp
• equal <-> equidistant

How the ideas connect

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

Deeper understanding

Go beyond memorising Chord Theorems at a Glance. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

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

Common mistakes

Exam focus

Core Idea

Key Rule

Second Step

Worked Method

Exam focus

Quick revision

• bisect <-> perp
• equal <-> equidistant
• d = 5 cm
• converse is separate
• CHORD THEOREMS AT A GLANCE: Keep the names and the one-line conclusion. Proofs live in the textbook; the exam often wants the st
• Unique circle: If 3 non-collinear points, then exactly one circumcircle.

Quote the matching converse. A theorem and its converse are two exam sentences, not one.

For long answers on Chord Theorems at a Glance: define clearly → explain the main parts → give one example → state one exam caution.