Kenneth ChengMathematical modeling
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Euclidean figures and circle theorems

Read the idea, work independently, then explain what changed.

2027 JM01 考試大綱 · 12. Rectilinear figures and circles · PDF 3 / printed page 3

TOPIC 01

Euclidean figures and circle theorems

Use similarity, angle and circle theorems with labeled points and compatible arcs.

What you will be able to explain

  • Use similarity, angle and circle theorems with labeled points and compatible arcs.
  • Justify the method and check the conditions in a new situation.

Model or definition

Use similarity, angle and circle theorems with labeled points and compatible arcs.

similarity length ratio k⇒area ratio k2\text{similarity length ratio }k\Rightarrow\text{area ratio }k^2

Conditions

Inscribed-angle comparisons must subtend the same arc on the stated side; cyclic opposite angles sum to 180°.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Do similar triangles with doubled sides have doubled areas?

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Circle x²+y²=4; line y=1. 2 intersections; chord length=3.4641. Tangency occurs exactly when h=r.

Circle x²+y²=4; line y=1. 2 intersections; chord length=3.4641. Tangency occurs exactly when h=r.

Explain: Compare two admissible cases and explain their different results using the stated model.

Transfer: Compare one theorem proof and a tempting use with the wrong corresponding arc.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Worked example

A convex polygon has t+3 sides. Find its interior-angle sum in degrees.

t=2t=2
  • Inscribed-angle comparisons must subtend the same arc on the stated side; cyclic opposite angles sum to 180°.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Identify the relevant geometric theorem and verify its hypotheses.
Hint 2
Use this intermediate relation.
S=(n−2)180∘S=(n-2)180^\circ
Worked solution

  1. Identify the relevant geometric theorem and verify its hypotheses.

  2. Apply the stated relation and retain its conditions.

    S=(5−2)180=540S=(5-2)180=540

  3. Triangulation from one vertex gives n−2 triangles.

The requested value is 540.

Checks and common pitfalls: Triangulation from one vertex gives n−2 triangles.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

A cyclic quadrilateral has one interior angle a°. Find its opposite interior angle.

a=23a=23
  • Inscribed-angle comparisons must subtend the same arc on the stated side; cyclic opposite angles sum to 180°.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Identify the relevant geometric theorem and verify its hypotheses.
Hint 2
Opposite angles sum to 180^\circ.
Worked solution

  1. Identify the relevant geometric theorem and verify its hypotheses.

  2. Apply the stated relation and retain its conditions.

    β=180−23=157∘\beta=180-23=157^\circ

  3. It is the opposite angle, not an adjacent angle.

The requested value is 157.

Checks and common pitfalls: It is the opposite angle, not an adjacent angle.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

A chord of a circle radius 5t is distance 3t from the center. Find its length.

t=4t=4
  • Inscribed-angle comparisons must subtend the same arc on the stated side; cyclic opposite angles sum to 180°.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Identify the relevant geometric theorem and verify its hypotheses.
Hint 2
The perpendicular from center bisects the chord.
Worked solution

  1. Identify the relevant geometric theorem and verify its hypotheses.

  2. Apply the stated relation and retain its conditions.

    (L/2)2=(20)2−(12)2=256(L/2)^2=(20)^2-(12)^2=256

  3. Apply the stated relation and retain its conditions.

    L=32=32L=32=32

  4. The right-triangle calculation gives half the chord first.

The requested value is 32.

Checks and common pitfalls: The right-triangle calculation gives half the chord first.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

A convex polygon has t+3 sides. Find its interior-angle sum in degrees.

t=5t=5
  • Inscribed-angle comparisons must subtend the same arc on the stated side; cyclic opposite angles sum to 180°.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Identify the relevant geometric theorem and verify its hypotheses.
Hint 2
Use this intermediate relation.
S=(n−2)180∘S=(n-2)180^\circ
Worked solution

  1. Identify the relevant geometric theorem and verify its hypotheses.

  2. Apply the stated relation and retain its conditions.

    S=(8−2)180=1080S=(8-2)180=1080

  3. Triangulation from one vertex gives n−2 triangles.

The requested value is 1080.

Checks and common pitfalls: Triangulation from one vertex gives n−2 triangles.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

A right triangle has legs 3t,4t. Find the hypotenuse.

t=6t=6
  • Inscribed-angle comparisons must subtend the same arc on the stated side; cyclic opposite angles sum to 180°.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Identify the relevant geometric theorem and verify its hypotheses.
Hint 2
Use this intermediate relation.
c2=(3t)2+(4t)2c^2=(3t)^2+(4t)^2
Worked solution

  1. Identify the relevant geometric theorem and verify its hypotheses.

  2. Apply the stated relation and retain its conditions.

    c=5(6)=30c=5(6)=30

  3. Pythagoras applies to the legs opposite the right angle.

The requested value is 30.

Checks and common pitfalls: Pythagoras applies to the legs opposite the right angle.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

Similar triangles have length ratio 2:3. The smaller area is 4t. Find the larger area.

t=7t=7
  • Inscribed-angle comparisons must subtend the same arc on the stated side; cyclic opposite angles sum to 180°.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Identify the relevant geometric theorem and verify its hypotheses.
Hint 2
Area ratio=(2/3)^2
Worked solution

  1. Identify the relevant geometric theorem and verify its hypotheses.

  2. Apply the stated relation and retain its conditions.

    Alarge=4(7)⋅9/4=63A_{\rm large}=4(7)\cdot9/4=63

  3. Area ratios square the corresponding length ratio.

The requested value is 63.

Checks and common pitfalls: Area ratios square the corresponding length ratio.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

A central angle subtends a minor arc of 2a°. Find an inscribed angle subtending that same arc.

a=28a=28
  • Inscribed-angle comparisons must subtend the same arc on the stated side; cyclic opposite angles sum to 180°.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Identify the relevant geometric theorem and verify its hypotheses.
Hint 2
Use this intermediate relation.
∠inscribed=∠central/2\angle_{\rm inscribed}=\angle_{\rm central}/2
Worked solution

  1. Identify the relevant geometric theorem and verify its hypotheses.

  2. Apply the stated relation and retain its conditions.

    ∠=56/2=28∘\angle=56/2=28^\circ

  3. The angle must subtend the stated same arc.

The requested value is 28.

Checks and common pitfalls: The angle must subtend the stated same arc.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

A central angle subtends a minor arc of 2a°. Find an inscribed angle subtending that same arc.

a=29a=29
  • Inscribed-angle comparisons must subtend the same arc on the stated side; cyclic opposite angles sum to 180°.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Identify the relevant geometric theorem and verify its hypotheses.
Hint 2
Use this intermediate relation.
∠inscribed=∠central/2\angle_{\rm inscribed}=\angle_{\rm central}/2
Worked solution

  1. Identify the relevant geometric theorem and verify its hypotheses.

  2. Apply the stated relation and retain its conditions.

    ∠=58/2=29∘\angle=58/2=29^\circ

  3. The angle must subtend the stated same arc.

The requested value is 29.

Checks and common pitfalls: The angle must subtend the stated same arc.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

A cyclic quadrilateral has one interior angle a°. Find its opposite interior angle.

a=30a=30
  • Inscribed-angle comparisons must subtend the same arc on the stated side; cyclic opposite angles sum to 180°.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Identify the relevant geometric theorem and verify its hypotheses.
Hint 2
Opposite angles sum to 180^\circ.
Worked solution

  1. Identify the relevant geometric theorem and verify its hypotheses.

  2. Apply the stated relation and retain its conditions.

    β=180−30=150∘\beta=180-30=150^\circ

  3. It is the opposite angle, not an adjacent angle.

The requested value is 150.

Checks and common pitfalls: It is the opposite angle, not an adjacent angle.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

A circle of radius t has a sector angle π/3 radians. Find its sector area.

t=11t=11
  • Inscribed-angle comparisons must subtend the same arc on the stated side; cyclic opposite angles sum to 180°.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Identify the relevant geometric theorem and verify its hypotheses.
Hint 2
Use this intermediate relation.
A=r2θ/2A=r^2\theta/2
Worked solution

  1. Identify the relevant geometric theorem and verify its hypotheses.

  2. Apply the stated relation and retain its conditions.

    A=121π/6A=121\pi/6

  3. The radian formula requires radians, and area scales with radius squared.

The requested value is 63.3554518474.

Checks and common pitfalls: The radian formula requires radians, and area scales with radius squared.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

A chord of a circle radius 5t is distance 3t from the center. Find its length.

t=12t=12
  • Inscribed-angle comparisons must subtend the same arc on the stated side; cyclic opposite angles sum to 180°.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Identify the relevant geometric theorem and verify its hypotheses.
Hint 2
The perpendicular from center bisects the chord.
Worked solution

  1. Identify the relevant geometric theorem and verify its hypotheses.

  2. Apply the stated relation and retain its conditions.

    (L/2)2=(60)2−(36)2=2304(L/2)^2=(60)^2-(36)^2=2304

  3. Apply the stated relation and retain its conditions.

    L=96=96L=96=96

  4. The right-triangle calculation gives half the chord first.

The requested value is 96.

Checks and common pitfalls: The right-triangle calculation gives half the chord first.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

A circle of radius t has a sector angle π/3 radians. Find its sector area.

t=13t=13
  • Inscribed-angle comparisons must subtend the same arc on the stated side; cyclic opposite angles sum to 180°.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Identify the relevant geometric theorem and verify its hypotheses.
Hint 2
Use this intermediate relation.
A=r2θ/2A=r^2\theta/2
Worked solution

  1. Identify the relevant geometric theorem and verify its hypotheses.

  2. Apply the stated relation and retain its conditions.

    A=169π/6A=169\pi/6

  3. The radian formula requires radians, and area scales with radius squared.

The requested value is 88.4881930761.

Checks and common pitfalls: The radian formula requires radians, and area scales with radius squared.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

A chord of a circle radius 5t is distance 3t from the center. Find its length.

t=14t=14
  • Inscribed-angle comparisons must subtend the same arc on the stated side; cyclic opposite angles sum to 180°.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Identify the relevant geometric theorem and verify its hypotheses.
Hint 2
The perpendicular from center bisects the chord.
Worked solution

  1. Identify the relevant geometric theorem and verify its hypotheses.

  2. Apply the stated relation and retain its conditions.

    (L/2)2=(70)2−(42)2=3136(L/2)^2=(70)^2-(42)^2=3136

  3. Apply the stated relation and retain its conditions.

    L=112=112L=112=112

  4. The right-triangle calculation gives half the chord first.

The requested value is 112.

Checks and common pitfalls: The right-triangle calculation gives half the chord first.

Think first. Reveal a hint when the class is ready.

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • Use similarity, angle and circle theorems with labeled points and compatible arcs.
    • Which condition is essential in euclidean figures and circle theorems?
    • Do similar triangles with doubled sides have doubled areas?

    Board plan

    • Model or definition: Use similarity, angle and circle theorems with labeled points and compatible arcs.
      similarity length ratio k⇒area ratio k2\text{similarity length ratio }k\Rightarrow\text{area ratio }k^2
    • Conditions: Inscribed-angle comparisons must subtend the same arc on the stated side; cyclic opposite angles sum to 180°.

    Anticipated thinking

    • Arc, chord and radius lengths are different quantities.

    Assessment checklist

    • 1 mark: choose the correct representation and conditions.
    • 1 mark: establish the intermediate relation.
    • 1 mark: complete a connected calculation or proof.
    • 1 mark: interpret and check the conclusion.

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    Curriculum and source notes ↗