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

46. Perimeters, and polygons divided into triangles

Goal: Your student can prove that perimeters of similar polygons are as their sides, and that similar polygons divide into similar triangles (and the converse).

⏱ 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)
  • A protractor

What's in the book

Propositions XXIV–XXVI (§§ 364–366) and Exercises 252–255.

The lesson, step by step

  1. Warm up

    Ask: State the theorem about altitudes of similar triangles.

  2. The new idea

    The perimeters of two similar polygons have the same ratio as any two homologous sides.

    If two polygons are similar, they are composed of the same number of triangles, similar each to each, and similarly placed.

    Prop. XXVI is the converse of Prop. XXV.

  3. Draw the figure

    Draw a pentagon and a larger similar one (double every side, copying each angle). From one vertex of each, draw the diagonals.

  4. Walk through the proof

    XXIV: the sides are in a series of equal ratios, so the sum of the antecedents is to the sum of the consequents as one side to its partner (§ 335).

    XXV: the first two triangles are similar by an angle and including sides. Subtract equal angles to get equal angles in the next pair, and continue.

    XXVI: add the equal angles of the triangles to get the polygon's angles, and chain the side ratios.

  5. 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.

  6. Exercises

    Ex. 254. AD and BE are two altitudes of the triangle CAB. Show that the triangles CED and CAB are similar.

    Ex. 255. If two circles are tangent to each other, the chords formed by a straight line drawn through the point of contact have the same ratio as the diameters of the circles.

  7. Wrap up

    Ask: Two similar triangles have sides in the ratio 2 : 3. What is the ratio of their perimeters? (2 : 3.)

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