Wentworth's Plane and Solid Geometry · Book III, Props. XXXIX–XLI · Book 4: Book III: Proportion. Similar Polygons

53. Mean proportional, extreme and mean ratio, similar polygons

Goal: Your student can construct a mean proportional, divide a line in extreme and mean ratio, and construct a polygon similar to a given one on a given side.

⏱ About 40 minutes · 7 steps

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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 XXXIX–XLI (§§ 388–391), the definition of extreme and mean ratio, and Exercise 289.

The lesson, step by step

  1. Warm up

    Ask: Construct a fourth proportional and explain it.

  2. The new idea

    To find the mean proportional between two given straight lines.

    To divide a given line in extreme and mean ratio.

    Upon a given line homologous to a given side of a given polygon, to construct a polygon similar to the given polygon.

  3. Draw the figure

    Prop. XXXIX: on a line lay off AC = 4 cm and CB = 9 cm. Draw a semicircle on AB and erect a perpendicular at C. Measure it: it should be 6 cm.

    Prop. XL: on a 10 cm line AB, follow the book's steps with a circle of radius half AB tangent at B.

  4. Walk through the proof

    XXXIX: the perpendicular from a point of the circumference to the diameter is the mean proportional between the segments (§ 370).

    XL: the tangent AB is the mean proportional between the whole secant AG and its outer part AF (§ 381). Rearranging the proportion shows AB × CB = AC².

    XLI: copy the triangles of the polygon one by one with equal angles; similar triangles similarly placed make similar polygons (§ 366).

  5. 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.

  6. Exercises

    Ex. 289. Construct x, if x = √(ab). Special cases : (1) a = 2, b = 3 ; (2) a = 1, b = 6 ; (3) a = 3, b = 7.

  7. Wrap up

    Measure the two parts of your line divided in extreme and mean ratio. For 10 cm they should be about 6.18 cm and 3.82 cm.

Tip

The division in extreme and mean ratio (often called the golden section today) returns in Book V for the regular decagon and pentagon.

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