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

52. Constructing proportional lines

Goal: Your student can divide a line into parts proportional to given lines, and construct a fourth proportional and a third proportional.

⏱ 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

Problems of Construction: Propositions XXXVI–XXXVIII (§§ 385–387) and Exercises 286–288.

The lesson, step by step

  1. Warm up

    Ask: How do we divide a line into equal parts? Why does it work?

  2. The new idea

    To divide a given straight line into parts proportional to any number of given lines.

    To find the fourth proportional to three given straight lines.

    To find the third proportional to two given straight lines.

  3. Draw the figure

    Your student does Ex. 286 as the drawing for Prop. XXXVI: draw a 12 cm line AB and a slanting line AX; on AX lay off 3, 6 and 7 equal steps; join the last point to B and draw parallels.

    Then construct a fourth proportional to lines of 3, 4 and 5 cm.

  4. Walk through the proof

    All three rest on Prop. XIII: a line parallel to one side of a triangle divides the other sides proportionally.

    XXXVII: on one side of an angle lay off m and n; on the other p; join and draw a parallel. Then m : n = p : DF.

    XXXVIII is the same with p = n.

  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. 286. Divide a line 12 inches long into three parts proportional to the numbers 3, 6, 7.

    Ex. 287. The square of the altitude of an equilateral triangle is equal to three fourths of the square of one side of the triangle.

  7. Wrap up

    Measure the fourth proportional you constructed and check: 3 : 4 = 5 : x gives x = 6⅔.

Answers to the book's problems

Ex. 286: 2¼ in, 4½ in, 5¼ in. Ex. 287: the altitude bisects the base, so h² = a² − (a/2)² = ¾a² (§ 372).

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

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