Wentworth's Plane and Solid Geometry · Book IV, Props. XVIII–XX · Book 5: Book IV: Areas of Polygons

63. Similar polygons of given size

Goal: Your student can construct a polygon similar to one polygon and equal to another, a square in a given ratio to a square, and a polygon similar to a given one in a given ratio.

⏱ 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 XVIII–XX (§§ 426–428) and Exercises 381–386.

The lesson, step by step

  1. Warm up

    Ask: Turn a triangle into a square, step by step.

  2. The new idea

    To construct a polygon similar to a given polygon P and equivalent to a given polygon Q.

    To construct a square which shall have a given ratio to a given square.

    To construct a polygon similar to a given polygon and having a given ratio to it.

  3. Draw the figure

    Prop. XIX: take AB as a side of the given square; lay off m and n on a slanting line; draw a parallel to extend AB to C so that AB : BC = m : n; then a semicircle on AC and a perpendicular at B.

  4. Walk through the proof

    XVIII: find squares equal to P and Q, with sides m and n; the fourth proportional to m, n and AB gives the side of the new polygon (§§ 386, 412).

    XIX: BD is the mean proportional between a and b, so a² : x² = a : b = m : n.

    XX: construct a side whose square has the given ratio (Prop. XIX), then build the similar polygon on it (Book III, Prop. XLI).

  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. 383. To construct a triangle equivalent to a given triangle, and having one side equal to a given length l.

    Ex. 384. To transform a triangle into an equivalent right triangle.

  7. Wrap up

    Ask your student to explain, without drawing, how to make a square half the size of a given square.

Answers to the book's problems

Ex. 381: add the squares two at a time by Prop. XI. Ex. 384: keep the base; erect a perpendicular at one end equal to the altitude, and join.

Worked out for this site from the scan. If a number in your copy differs, trust the book.

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