Kenneth ChengMathematical modeling
← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

Intersections and distance formulas

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

高二選擇性必修 第一册(A版).pdf · 2.3 · PDF 75 / printed page 70

Revisit first: Equations of a line

TOPIC 01

Intersections and distance formulas

Solve intersections and calculate point-line and parallel-line distances with geometric checks.

What you will be able to explain

  • Solve intersections and calculate point-line and parallel-line distances with geometric checks.
  • Justify the method and check the conditions in a new situation.

Intersection

Solve both linear equations simultaneously and check both.

Normalized distance

The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.

d=∣Ax0+By0+C∣/A2+B2d=|Ax_0+By_0+C|/\sqrt{A^2+B^2}

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Why does multiplying a line equation by 100 leave its distance formula unchanged?

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

y=1x+1. Angle is measured modulo 180°; distance from the origin=0.7071. This slope form excludes vertical lines.

y=1x+1. Angle is measured modulo 180°; distance from the origin=0.7071. This slope form excludes vertical lines.

Explain: Calculate two valid cases and explain the change using the defining relation.

Transfer: Test invariance under positive and negative scaling of the equation.

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

Find the x-coordinate of the intersection.

x+y=4,x−y=2x+y=4,\quad x-y=2
  • The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
2x=2t+22x=2t+2
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    2x=6⇒x=32x=6\Rightarrow x=3

  3. Adding eliminates y; subtraction finds y if needed.

The requested value is 3.

Checks and common pitfalls: Adding eliminates y; subtraction finds y if needed.

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

02 / Standard#Worked example

Find the foot of the perpendicular from P to the line.

P=(3,2),y=xP=(3,2),\quad y=x
  • The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
H=(h,h),(P−H)⋅(1,1)=0H=(h,h),\quad(P-H)\cdot(1,1)=0
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    3+2−2h=03+2-2h=0

  3. Apply the stated relation and retain its conditions.

    h=2.5h=2.5

  4. A foot lies on the line and its displacement from P is perpendicular to the line.

The requested relation or conclusion is shown below.

H=(2.5,2.5)H=(2.5,2.5)

Checks and common pitfalls: A foot lies on the line and its displacement from P is perpendicular to the line.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

03 / Transfer#Worked example

A=(0,1), B=(t,3), P is on y=0. Find the minimum of AP+PB.

t=4t=4
  • The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Reflect A to A\prime=(0,-1).
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    AP=A′PAP=A'P

  3. Apply the stated relation and retain its conditions.

    AP+PB≥A′B=42+42AP+PB\ge A'B=\sqrt{4^2+4^2}

  4. The segment A′B crosses y=0, so equality is attainable.

The requested value is 5.65685424949.

Checks and common pitfalls: The segment A′B crosses y=0, so equality is attainable.

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

04 / Foundation#Your turn

Find the x-coordinate of the intersection.

x+y=10,x−y=2x+y=10,\quad x-y=2
  • The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
2x=2t+22x=2t+2
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    2x=12⇒x=62x=12\Rightarrow x=6

  3. Adding eliminates y; subtraction finds y if needed.

The requested value is 6.

Checks and common pitfalls: Adding eliminates y; subtraction finds y if needed.

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

05 / Foundation#Your turn

Find the distance from (0,0) to x=t.

t=6t=6
  • The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
d=∣0−t∣d=|0-t|
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    d=6d=6

  3. The shortest segment is horizontal.

The requested value is 6.

Checks and common pitfalls: The shortest segment is horizontal.

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

06 / Foundation#Your turn

Find the distance from P to the line.

P=(0,0),3x+4y−35=0P=(0,0),\quad 3x+4y-35=0
  • The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
d=∣C∣/5d=|C|/5
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    d=35/9+16=7d=35/\sqrt{9+16}=7

  3. The normal (3,4) has length 5.

The requested value is 7.

Checks and common pitfalls: The normal (3,4) has length 5.

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

07 / Foundation#Your turn

Find the distance between the parallel lines.

3x+4y=0,6x+8y=803x+4y=0,\quad 6x+8y=80
  • The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
3x+4y=5t3x+4y=5t
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    d=∣40−0∣/5=8d=|40-0|/5=8

  3. Divide the second equation by two before comparing constants.

The requested value is 8.

Checks and common pitfalls: Divide the second equation by two before comparing constants.

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

08 / Standard#Your turn

Find the distance between the parallel lines.

3x+4y=0,6x+8y=903x+4y=0,\quad 6x+8y=90
  • The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
3x+4y=5t3x+4y=5t
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    d=∣45−0∣/5=9d=|45-0|/5=9

  3. Divide the second equation by two before comparing constants.

The requested value is 9.

Checks and common pitfalls: Divide the second equation by two before comparing constants.

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

09 / Standard#Your turn

Find the foot of the perpendicular from P to the line.

P=(10,2),y=xP=(10,2),\quad y=x
  • The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
H=(h,h),(P−H)⋅(1,1)=0H=(h,h),\quad(P-H)\cdot(1,1)=0
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    10+2−2h=010+2-2h=0

  3. Apply the stated relation and retain its conditions.

    h=6h=6

  4. A foot lies on the line and its displacement from P is perpendicular to the line.

The requested relation or conclusion is shown below.

H=(6,6)H=(6,6)

Checks and common pitfalls: A foot lies on the line and its displacement from P is perpendicular to the line.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

10 / Standard#Your turn

Reflect P in the line y=x.

P=(11,2)P=(11,2)
  • The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
P+P′=2HP+P\prime=2H
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    H=(6.5,6.5)H=(6.5,6.5)

  3. Apply the stated relation and retain its conditions.

    P′=2H−P=(2,11)P'=2H-P=(2,11)

  4. Reflection across y=x swaps coordinates.

The requested relation or conclusion is shown below.

P′=(2,11)P'=(2,11)

Checks and common pitfalls: Reflection across y=x swaps coordinates.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

11 / Standard#Your turn

A=(0,1), B=(t,3), P is on y=0. Find the minimum of AP+PB.

t=12t=12
  • The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Reflect A to A\prime=(0,-1).
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    AP=A′PAP=A'P

  3. Apply the stated relation and retain its conditions.

    AP+PB≥A′B=122+42AP+PB\ge A'B=\sqrt{12^2+4^2}

  4. The segment A′B crosses y=0, so equality is attainable.

The requested value is 12.6491106407.

Checks and common pitfalls: The segment A′B crosses y=0, so equality is attainable.

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

12 / Transfer#Your turn

Reflect P in the line y=x.

P=(13,2)P=(13,2)
  • The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
P+P′=2HP+P\prime=2H
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    H=(7.5,7.5)H=(7.5,7.5)

  3. Apply the stated relation and retain its conditions.

    P′=2H−P=(2,13)P'=2H-P=(2,13)

  4. Reflection across y=x swaps coordinates.

The requested relation or conclusion is shown below.

P′=(2,13)P'=(2,13)

Checks and common pitfalls: Reflection across y=x swaps coordinates.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

13 / Transfer#Your turn

A=(0,1), B=(t,3), P is on y=0. Find the minimum of AP+PB.

t=14t=14
  • The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Reflect A to A\prime=(0,-1).
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    AP=A′PAP=A'P

  3. Apply the stated relation and retain its conditions.

    AP+PB≥A′B=142+42AP+PB\ge A'B=\sqrt{14^2+4^2}

  4. The segment A′B crosses y=0, so equality is attainable.

The requested value is 14.5602197786.

Checks and common pitfalls: The segment A′B crosses y=0, so equality is attainable.

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

Focus on one question

End-of-lesson check and correction

Review your latest checked answers and explanations. A draft change requires a fresh check. Written work needs your self-assessment or a teacher’s review.

Enable JavaScript for a summary of your local work.

    Choose a foundation skill to revisit ↗

    Teacher preparation and assessment

    Question sequence

    • Solve intersections and calculate point-line and parallel-line distances with geometric checks.
    • Which condition is essential in intersections and distance formulas?
    • Why does multiplying a line equation by 100 leave its distance formula unchanged?

    Board plan

    • Intersection: Solve both linear equations simultaneously and check both.
    • Normalized distance: The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.
      d=∣Ax0+By0+C∣/A2+B2d=|Ax_0+By_0+C|/\sqrt{A^2+B^2}

    Anticipated thinking

    • Subtracting constants before matching parallel-line coefficients gives a wrong distance.

    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.

    No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

    Curriculum and source notes ↗