Wentworth's Plane and Solid Geometry · Book VI, Props. XXIII–XXV · Book 7: Book VI: Lines and Planes in Space
89. Projections, the angle of a line with a plane, and the common perpendicular
Goal: Your student can prove that the projection of a line on a plane is a line, define the inclination of a line to a plane, and prove that two skew lines have exactly one common perpendicular.
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)
- A calculator (optional)
What's in the book
Propositions XXIII–XXV (§§ 563–568), with the definitions of projecting plane and inclination, and Exercises 613–615.
The lesson, step by step
Warm up
Ask: What is the projection of a point on a plane?
The new idea
The projection of a straight line not perpendicular to a plane upon that plane is a straight line.
The acute angle which a straight line makes with its projection upon a plane is the least angle which it makes with any line of the plane.
Between two straight lines not in the same plane, there can be one common perpendicular, and only one.
Draw the figure
Hold two pencils so that they are not parallel and do not meet (skew lines). Find with a third pencil the one direction that meets both at right angles.
For Prop. XXIV, draw the line BA meeting the plane at A, its projection AC, and another line AD in the plane with AD = AC.
Walk through the proof
XXIII: the plane through AB and a perpendicular HK is perpendicular to MN and contains all the perpendiculars from AB (§ 553); its meeting with MN is a straight line (§ 506).
XXIV: in triangles BAC and BAD, two sides are equal but BC < BD (the perpendicular is shortest), so angle BAC < angle BAD (§ 155).
XXV: pass a plane MN through AB parallel to DC, and a plane through DC perpendicular to MN; from the point where it crosses AB, the perpendicular to MN meets DC at right angles. Any other common perpendicular would give two perpendiculars from one point to MN.
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.
Exercises
Ex. 613. From a point A, 4 inches from a plane MN, an oblique line AC 5 inches long is drawn to the plane and made to turn around the perpendicular AB dropped from A to the plane. Find the area of the circle described by the point C.
Ex. 614. From a point A, 8 inches from a plane MN, a perpendicular AB is drawn to the plane; with B as centre, and a radius equal to 6 inches, a circle is described in the plane; at any point C of this circumference a tangent CD 24 inches long is drawn. Find the distance from A to D.
Wrap up
Ask: Ex. 615: when is the projection of a line as long as the line itself? (When the line is parallel to the plane.)
Answers to the book's problems
Ex. 613: BC = √(25 − 16) = 3 in, so the area is 9π, about 28.27 sq in. Ex. 614: AC = √(64 + 36) = 10 in; CD is perpendicular to the radius BC and so to AC (§ 518); AD = √(100 + 576) = 26 in. Ex. 615: the line is parallel to the plane (or lies in it).
Worked out for this site from the scan. If a number in your copy differs, trust the book.