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

48. The right triangle and the Pythagorean theorem

Goal: Your student can prove the mean-proportional relations in a right triangle and the theorem that the squares of the legs add to the square of the hypotenuse.

⏱ 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

Numerical Properties of Figures: Propositions XXVII and XXVIII (§§ 367–374), with corollaries, including that the diagonal and side of a square are incommensurable, and the definition of projection.

The lesson, step by step

  1. Warm up

    Ask: What is the mean proportional between 4 and 25?

  2. The new idea

    Prop. XXVII (§ 367) has three parts: the perpendicular from the right angle to the hypotenuse makes triangles similar to the whole and to each other; the perpendicular is the mean proportional between the segments of the hypotenuse; each leg is the mean proportional between the hypotenuse and its adjacent segment. Then:

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

  3. Draw the figure

    Draw a right triangle ABC with the right angle at C (legs 6 cm and 8 cm). Drop the perpendicular CF to the hypotenuse AB.

  4. Walk through the proof

    XXVII: each small right triangle shares an acute angle with the big one, so all three are similar (§ 356). Write the proportions from the similar triangles.

    XXVIII: from part 3, AC² = AB × AF and CB² = AB × BF. Add: AC² + CB² = AB × (AF + BF) = AB².

  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

    Check with numbers: 6² + 8² = 36 + 64 = 100 = 10². Measure the hypotenuse of your drawing.

  7. Wrap up

    Read § 373: the diagonal of a square is the side times √2, so they are incommensurable. Then read § 374 on projections, which are needed next time.

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