Completing a Circle Without the Centre (named)

Completing a Circle Without the Centre (named)

What is it?

One school method steps congruent triangles / equal chords around the arc until it closes. Another uses 120 deg interior angles so a regular hexagon appears (equal chords). These are named methods in the practical-geometry chapter: equal chords, not a freehand guess.

In simple words

• One school method steps congruent triangles / equal chords around the arc until it closes.
• CONGRUENT TRIANGLES: Copy a chord length repeatedly along the remaining gap.
• 120 DEG / HEXAGON: Regular hexagon interior 120 deg.
• Worked-method reminder: = 120 deg

Important terms

COMPLETING A CIRCLE WITHOUT THE CENTRE (NAMED): One school method steps congruent triangles / equal chords around the arc until it closes. Another uses 120 deg interior angles so a regular hexagon appears (equal chords). These are named methods in the practical-geometry chapter: equal chords, not a freehand guess.

CONGRUENT TRIANGLES: Copy a chord length repeatedly along the remaining gap. Congruent triangles keep the same radius without naming O at first. Join the new vertices to close the circumference.

120 DEG / HEXAGON: Regular hexagon interior 120 deg. Building with that angle and equal sides completes the circle in six equal chords. Interior is 120, not 60 (60 is the central angle of the hexagon).

What you should learn

• Explain completing a circle without the centre (named) using the correct definitions, conditions, and notation.
• Apply equal chords around, hexagon 120 deg method correctly in routine and exam-style questions.
• Show each important step, simplify the result, and check that it satisfies the original conditions.

Examples

• Copy chord length
• Step around
• Not random radii

CONGRUENT TRIANGLES

Copy a chord length repeatedly along the remaining gap. Congruent triangles keep the same radius without naming O at first. Join the new vertices to close the circumference.

120 DEG / HEXAGON

Regular hexagon interior 120 deg. Building with that angle and equal sides completes the circle in six equal chords. Interior is 120, not 60 (60 is the central angle of the hexagon).

Full school-level explanation

One school method steps congruent triangles / equal chords around the arc until it closes. Another uses 120 deg interior angles so a regular hexagon appears (equal chords). These are named methods in the practical-geometry chapter: equal chords, not a freehand guess.

Equal steps on the rim:

WHAT — COMPLETING A CIRCLE WITHOUT THE CENTRE (NAMED)

One school method steps congruent triangles / equal chords around the arc until it closes. Another uses 120 deg interior angles so a regular hexagon appears (equal chords). These are named methods in the practical-geometry chapter: equal chords, not a freehand guess. The symbols and conditions are part of the mathematics, so include them when stating the rule.

WHY — Why this topic matters

Completing A Circle Without The Centre (Named) matters because accurate compass-and-straightedge constructions apply circle theorems in a visible way. Each construction works because a locus or perpendicular condition fixes the required centre, circle, or tangent.

HOW — How to solve and explain it

Start by identifying what is given and what must be found. Work through CONGRUENT TRIANGLES, then 120 DEG / HEXAGON. Worked: interior angle of a regular hexagon used in the 120 deg method. regular hexagon has 6 sides interior angle = [(6-2)*180]/6 = 7206 = 120 deg Finish by checking signs, restrictions, units, and whether the answer matches the question.

CONGRUENT TRIANGLES

Copy a chord length repeatedly along the remaining gap. Congruent triangles keep the same radius without naming O at first. Join the new vertices to close the circumference.

Remember

• equal chords

Examples

• Copy chord length
• Step around
• Not random radii

120 DEG / HEXAGON

Regular hexagon interior 120 deg. Building with that angle and equal sides completes the circle in six equal chords. Interior is 120, not 60 (60 is the central angle of the hexagon).

Remember

• interior 120

Examples

• Regular hexagon
• Six equal chords
• Central angle of hexagon is 60

Remember

• equal chords around
• hexagon 120 deg method
• 120 deg
• interior 120, central 60
• equal chords
• interior 120
• equal chords around
• hexagon 120 deg method

How the ideas connect

Worked: interior angle of a regular hexagon used in the 120 deg method. regular hexagon has 6 sides interior angle = [(6-2)*180]/6 = 7206 = 120 deg

Deeper understanding

Go beyond memorising Completing a Circle Without the Centre (named). Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked: interior angle of a regular hexagon used in the 120 deg method. regular hexagon has 6 sides interior angle = [(6-2)*180]/6 = 7206 = 120 deg

Common mistakes

Exam focus

Core Idea

CONGRUENT TRIANGLES

120 DEG / HEXAGON

Worked Method

Exam focus

Quick revision

• equal chords around
• hexagon 120 deg method
• 120 deg
• interior 120, central 60
• COMPLETING A CIRCLE WITHOUT THE CENTRE (NAMED): One school method steps congruent triangles / equal chords around the arc until it closes. Another u
• CONGRUENT TRIANGLES: Copy a chord length repeatedly along the remaining gap. Congruent triangles keep the same radius wit

60 deg is the angle at the centre between adjacent hexagon radii, not the interior angle of the hexagon.

For long answers on Completing a Circle Without the Centre (named): define clearly → explain the main parts → give one example → state one exam caution.