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

49. Squares of sides in any triangle; medians

Goal: Your student can prove the formulas for the square of the side opposite an acute or obtuse angle, and the theorem on the median.

⏱ About 40 minutes · 6 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 XXIX–XXXI (§§ 375–377).

The lesson, step by step

  1. Warm up

    Ask: Prove the Pythagorean theorem. What is the projection of one line on another?

  2. The new idea

    In any triangle, the square of the side opposite an acute angle is equal to the sum of the squares of the other two sides diminished by twice the product of one of those sides by the projection of the other upon that side.

    In any obtuse triangle, the square of the side opposite the obtuse angle is equal to the sum of the squares of the other two sides increased by twice the product of one of those sides by the projection of the other upon that side.

    Prop. XXXI (§ 377) gives the sum and the difference of the squares of two sides in terms of the median to the third side.

  3. Draw the figure

    Draw an acute-angled triangle ABC and the altitude AD to BC, so DC is the projection of AC on BC. Draw an obtuse triangle too, with the altitude falling outside.

  4. Walk through the proof

    XXIX: DB = BC − DC (or DC − BC). Square it, add AD² to both sides, and use the Pythagorean theorem in the two right triangles.

    XXX: the same, with DB = BC + DC.

    XXXI: apply XXX to one half-triangle and XXIX to the other (the median makes one angle obtuse and one acute), then add and subtract.

  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. Wrap up

    Read Note 1 on page 164: by the Principle of Continuity, XXVIII, XXIX and XXX are one theorem. Ask your student to explain why (when the angle is right, the projection is zero).

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