LEARN · EXPLAIN · REVISE

Prove that a line with two distinct points in a plane lies entirely in the plane.

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

高一必修第二册(A版).pdf · 8.4 · PDF 131 / printed page 124

TOPIC 01

Positions of points, lines and planes in space

Build understanding of positions of points, lines and planes in space through definitions, contrasting cases and justified applications.

Use definitions, valid spatial reasoning and metric checks for positions of points, lines and planes in space.
Connect representations and justify the steps, including boundary cases.
Explain a solution and apply the idea to a changed situation.

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

01 / Standard#Your turn

A,B\in\alpha,\quad A\ne B

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use precise incidence conditions; diagrams alone do not prove spatial relationships.

Hint 2

Use the axiom with both points, not just one.

Worked solution

Use precise incidence conditions; diagrams alone do not prove spatial relationships.
Calculate or simplify this relation.
$A,B\in\alpha\Rightarrow AB\subseteq\alpha$
One common point permits a line to pass through the plane without lying in it.

The line AB is contained in α by the plane incidence axiom.

Checks and common pitfalls: One common point permits a line to pass through the plane without lying in it.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

What must be true before using the main rule for positions of points, lines and planes in space?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Use definitions, valid spatial reasoning and metric checks for positions of points, lines and planes in space.
Plane determination: Three noncollinear points determine one plane.
Line positions: Spatial lines may intersect, be parallel, or be skew.
Close with: conditions → representation → reasoning → check.