Wentworth's Plane and Solid Geometry · Book II, Props. XXI–XXV · Book 3: Book II: The Circle
32. First constructions: perpendiculars and bisecting
Goal: Your student can drop and erect perpendiculars, and bisect a line, an arc and an angle with ruler and compasses, and prove each construction correct.
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
The first Problems of Construction, Propositions XXI–XXV (§§ 300–304), and Exercises 134–137.
The lesson, step by step
Warm up
Ask: What is the locus of points equidistant from the ends of a line?
The new idea
From now on some propositions are problems: things to construct.
To let fall a perpendicular upon a given line from a given external point.
At a given point in a straight line, to erect a perpendicular to that line.
To bisect a given straight line.
To bisect a given arc.
To bisect a given angle.
Draw the figure
Your student does each construction, following the book: for Prop. XXIII, from A and B as centres, with equal radii greater than half AB, draw arcs crossing above and below the line, and join the crossings.
Read the note on page 112: given lines are drawn solid, resulting lines long-dashed, and helping lines short-dashed.
Walk through the proof
Each construction makes two points that are each equidistant from two other points. Two such points determine the perpendicular bisector (§ 161). That one idea proves Props. XXI, XXIII, XXIV and XXV.
Prop. XXII has a second method: the angle at B is inscribed in a semicircle, so it is a right angle (§ 290).
Your student proves it back
Close the book. Your student redraws the figure from memory, letters it the same way, and does each construction and explains why it works: 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
Try Exercises 134–137 on page 115:
Ex. 134. To construct an angle of 45° ; of 135°.
Ex. 137. To trisect a right angle.
Wrap up
Check the bisected angle with a protractor. A good construction should be accurate to a degree.
Tip
Keep the compass point sharp and the pencil sharp. Neat constructions are part of the lesson.
Answers to the book's problems
Ex. 134: erect a perpendicular and bisect the right angle (45°); 135° is a right angle plus 45°. Ex. 135: from each end of the side, with radius equal to the side, draw arcs; they meet at the third vertex. Ex. 136: an angle of the equilateral triangle is 60°; 150° = 90° + 60°. Ex. 137: on each arm of the right angle construct an angle of 60° (from an equilateral triangle, Ex. 135), turned toward the other arm; the two new lines cut the right angle into three angles of 30°.
Worked out for this site from the scan. If a number in your copy differs, trust the book.