Wentworth's Plane and Solid Geometry · Book I, Props. V–VII · Book 2: Book I: Rectilinear Figures
6. Perpendiculars: equal obliques and the shortest line
Goal: Your student can prove that equal obliques cut off equal segments, that only one perpendicular can be drawn from an outside point, and that the perpendicular is the shortest line to a line.
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 protractor
What's in the book
Propositions V, VI and VII (§§ 95–99), with Exercise 4 about the angle between clock hands.
The lesson, step by step
Warm up
Ask: State Prop. IV and its reason. What is a perpendicular?
The new idea
Three facts about a perpendicular to a line:
Only one perpendicular can be drawn to a given line from a given external point.
The perpendicular is the shortest line that can be drawn to a straight line from an external point.
Prop. V (§ 95) says that two lines from a point in the perpendicular, cutting off equal segments from its foot, are equal and make equal angles with the perpendicular.
Draw the figure
Draw a horizontal line AB. Mark P well above it and drop a perpendicular PC (use the corner of a set square or book). Draw another line PD from P to AB.
For Prop. VII, extend PC below AB to P', making CP' = PC, and join DP'.
Walk through the proof
Prop. V: fold the figure over the perpendicular. One side falls on the other, the equal segments match, so the two slanting lines coincide.
Prop. VI: extend PC to P' with CP' = PC. PCP' is a straight line, so PDP' is not. Angle PDC is half of angle PDP', which is not a straight angle, so PDC is not a right angle.
Prop. VII: PD = DP' (Prop. V). PC + CP' is a straight line, so it is shorter than PD + DP'. Halve both: PC < PD.
Your student proves it back
Close the book. Your student redraws the figure from memory, letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason ("because the radii are equal", "by § 143"). Point to each line or angle as it is named.
If a step is missing, ask "How do we know that?" rather than supplying it.
Exercises
Ex. 4 on page 19:
Ex. 4. Find the number of degrees in the angle included by the hands of a clock at 1 o'clock. 3 o'clock. 5 o'clock. 6 o'clock.
Hint: the hour marks are 360° ÷ 12 = 30° apart.
Wrap up
Your student explains why a ladder leaning against a wall is longer than the height it reaches.
Answers to the book's problems
Ex. 4: 30°, 90°, 150°, 180°.
Worked out for this site from the scan. If a number in your copy differs, trust the book.