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

59. Similar figures and the square on the hypotenuse

Goal: Your student can prove that similar triangles and polygons are as the squares of their sides, and Euclid's proof that the square on the hypotenuse equals the sum of the squares on the legs.

⏱ 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)
  • Colored pencils

What's in the book

Comparison of Polygons: Propositions VIII–X (§§ 411–416), and Exercise 355.

The lesson, step by step

  1. Warm up

    Ask: State the area of a triangle, and the theorem on homologous altitudes of similar triangles.

  2. The new idea

    The areas of two similar triangles are to each other as the squares of any two homologous sides.

    The areas of two similar polygons are to each other as the squares of any two homologous sides.

    The square on the hypotenuse of a right triangle is equivalent to the sum of the squares on the two legs.

  3. Draw the figure

    For Prop. X draw a right triangle ABC (right angle at A) with legs about 6 and 8 cm, and build a square outward on each side. Through A draw AL parallel to the side of the big square, and draw AD and CF as in the book.

  4. Walk through the proof

    VIII: the areas are as base × altitude, and in similar triangles the altitudes are in the same ratio as the bases.

    IX: divide both polygons into similar triangles from one vertex and add the ratios (§ 335).

    X: triangles ABD and FBC are equal (two sides and the included angle). The rectangle BL is double triangle ABD, and the square AF is double triangle FBC, so they are equivalent. In the same way rectangle CL equals square CH.

  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.

    Prop. X is long. Allow two sittings if needed, and let your student color the pairs of equal triangles.

  6. Exercises

    Ex. 355. Prove this proposition by § 410.

    (This refers to Prop. VIII.)

  7. Wrap up

    Ask: If one triangle has sides twice as long as a similar triangle, how many times its area is it? (4.)

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