Wentworth's Plane and Solid Geometry · Book II, Props. V–VIII · Book 3: Book II: The Circle
24. Chords and their distance from the centre
Goal: Your student can prove that a diameter perpendicular to a chord bisects it, and the theorems relating the length of a chord to its distance from the centre.
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)
What's in the book
Propositions V–VIII (§§ 245–252) with corollaries, and Exercise 91.
The lesson, step by step
Warm up
Ask: State Props. II and III from last time.
The new idea
A diameter perpendicular to a chord bisects the chord and the arcs subtended by it.
In the same circle or in equal circles, equal chords are equally distant from the centre. Conversely: Chords equally distant from the centre are equal.
In the same circle or in equal circles, if two chords are unequal, they are unequally distant from the centre; and the greater chord is at the less distance.
Prop. VIII (§ 251) is the converse of Prop. VII.
Draw the figure
Draw a circle with centre O and a chord AB. Draw a diameter perpendicular to AB, crossing it at M. Join OA and OB.
For Prop. VII, draw a longer chord and a shorter chord and the perpendiculars from O to each.
Walk through the proof
V: right triangles OAM and OBM have the hypotenuses (radii) equal and OM common, so they are equal (Prop. XXV of Book I): AM = BM, and the central angles are equal, so the arcs are equal.
VI: the perpendiculars bisect the equal chords, so the half-chords are equal, and the right triangles with radii as hypotenuses are equal.
VII: lay off a chord AG equal to the shorter one, from the same end A. Compare angles in the triangle formed by the midpoints, and use the unequal-sides theorems of Book I.
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. 91. The perpendicular bisectors of the sides of an inscribed polygon are concurrent (pass through the same point).
Wrap up
Ask: In a circle, which chord is the longest of all? (A diameter, § 252.)
Answers to the book's problems
Ex. 91: each side is a chord, and the perpendicular bisector of a chord passes through the centre (§ 248); so all of them pass through the centre.
Worked out for this site from the scan. If a number in your copy differs, trust the book.