Wentworth's Plane and Solid Geometry · Book VI, Props. II–IV · Book 7: Book VI: Lines and Planes in Space
81. A line perpendicular to a plane
Goal: Your student can prove that a line perpendicular to two lines of a plane at their meeting point is perpendicular to the plane, and that there is exactly one perpendicular through a point.
You'll need
- The book, open to these pages
- A ruler (straightedge) and a pair of compasses
- Sharp pencil, eraser and plain paper (or a geometry notebook)
- Two or three stiff cards (to stand for planes) and a few pencils or straws (to stand for lines)
What's in the book
Propositions II–IV (§§ 507–513), with the corollaries and the definition of the distance from a point to a plane.
The lesson, step by step
Warm up
Ask: What determines a plane? Name the five ways.
The new idea
If a straight line is perpendicular to each of two other straight lines at their point of intersection, it is perpendicular to the plane of the two lines.
All the perpendiculars that can be drawn to a straight line at a given point lie in a plane which is perpendicular to the line at the given point.
Through a given point there can be one perpendicular to a given plane, and only one.
§ 512: the perpendicular is the shortest line from a point to a plane, and its length is the distance (§ 513).
Draw the figure
Stand a pencil upright on the table and lay two other pencils flat, crossing at its foot. Check with the corner of a card that the upright pencil makes right angles with both.
For Prop. II, draw plane MN with lines BC and BD through B, the line AB up from B, and AB extended below the plane to F with BF = AB.
Walk through the proof
II: BC and BD are each perpendicular to AF at its middle point, so A and F are equally distant from C and from D (§ 160). Then triangles ACD and FCD are equal, which gives equal angles at C, then equal triangles ACE and FCE. So AE = FE, and BE (any other line through B) is also perpendicular to AF.
III: if a perpendicular BE were outside the plane MN, the plane of AB and BE would contain two perpendiculars to AB at B, which is impossible (§ 83).
IV: build the perpendicular by passing a plane through A perpendicular to a line of MN; then show any other line from A is oblique.
Your student proves it back
Close the book. Your student sketches the figure from memory (a simple perspective drawing is fine), letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason ("by § 501", "by § 507"). Point to each line and plane as it is named.
If a step is missing, ask "How do we know that?" rather than supplying it.
Wrap up
Ask: Why does a carpenter check that a post is upright by testing it against two different directions, not just one? (Prop. II.)