LEARN · EXPLAIN · REVISE

How many planes are determined by three noncollinear points?

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 / Foundation#Worked example

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

The noncollinear condition is essential.

Worked solution

Use precise incidence conditions; diagrams alone do not prove spatial relationships.
Calculate or simplify this relation.
$A,B,C\text{ noncollinear }\Rightarrow\exists!\alpha:A,B,C\in\alpha$
Three collinear points instead lie in infinitely many planes.

The requested value is 1.

Checks and common pitfalls: Three collinear points instead lie in infinitely many planes.

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.