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D,E are midpoints of AB,AC. Choose directed DE.

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高一必修 第二册(A版).pdf · 6.4 · PDF 45 / printed page 38

Revisit first: Coordinates, bases and divisionDot products and constrained projection

TOPIC 01

Vector proofs in triangles

Turn geometric hypotheses into vector equations; prove midpoint, centroid and orthocenter properties with their assumptions visible.

What you will be able to explain

  • Choose a convenient origin without changing a geometric claim.
  • Derive a median identity from a squared norm.
  • Separate a theorem, its converse and nondegeneracy conditions.

Try it. Leave your reasoning visible.

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

01 / Foundation#Your turn

D,E are midpoints of AB,AC. Choose directed DE.

AB=u, AC=vAB=u,\ AC=v
Skills and prerequisite lessons

Scope: JM02

  1. Option A(v−u)/2(v-u)/2
  2. Option B(u+v)/2(u+v)/2
  3. Option Cv−uv-u
  4. Option D(u−v)/2(u-v)/2
Working and explanation
BUILD THE REASONING
Hint 1
D has position u/2.
Hint 2
E−D gives the required direction.
Worked solution
  1. Subtract midpoint positions.
    DE=v/2−u/2DE=v/2-u/2
  2. Factor one half.
    DE=(v−u)/2DE=(v-u)/2
A.
Checks and common pitfalls: A diagram does not determine a directed sign without an endpoint order.
Think first. Reveal a hint when the class is ready.
Focus on one question
Teacher preparation and assessment
40-minute lesson
MinutesFocusTeaching and student action
5Prediction and diagnosisCompare one numerical triangle with a universal claim.
8Concept derivationDerive midpoint and median identities with A as origin.
10Guided applicationUse dot products to verify right angles and an orthocenter.
12Transfer and explanationProve the rhombus converse and discuss the missing assumption.
5Exit evidenceRepair a one-line proof with an unjustified conclusion.
Question sequence
  • Which origin makes two vertex positions simplest?
  • Where exactly did the parallelogram assumption enter?
Board development
  • Arrange givens, vector translation, identity and conclusion in four lines.
  • For the orthocenter, write one dot-product equation for each altitude and verify the third.
Expected correct and incorrect responses
  • Students say the diagonals are perpendicular because their lengths match. Expand their dot product to reveal the actual side-length condition.
  • Correct: “The parallelogram assumption makes its diagonal vectors u+v and v−u.”
  • Incorrect: “Every quadrilateral with perpendicular diagonals is a rhombus.”
Follow-up questions
  • Can the centroid formula still be evaluated for collinear points, and does it create a nondegenerate triangle?
Remediation
  • Start with the midpoint subtraction E−D before introducing a length formula.
Exit check
  • Write the missing hypothesis in “perpendicular diagonals imply a rhombus”.
Assessment checklist and sample responses
  • Look for a justified origin, correct directed vectors and a return to the geometric claim. Sample: (u+v)·(u−v)=|u|²−|v|², so perpendicular diagonals force equal adjacent sides in a parallelogram.
    Correct: “The parallelogram assumption makes its diagonal vectors u+v and v−u.”
No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.
Curriculum and source notes ↗