Wentworth's Plane and Solid Geometry · Book I, Props. VIII–IX · Book 2: Book I: Rectilinear Figures

7. Lines drawn to a line: which is longer?

Goal: Your student can prove that the lines from a point to the ends of a segment are longer than lines from an inside point, and that the more remote oblique is the longer.

⏱ About 35 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 VIII and IX (§§ 100–102) with the corollary that only two equal lines can be drawn from a point to a straight line.

The lesson, step by step

  1. Warm up

    Ask: Prove that the perpendicular is the shortest line from a point to a line.

  2. The new idea

    The sum of two lines drawn from a point to the extremities of a straight line is greater than the sum of two other lines similarly drawn, but included by them.

    Of two straight lines drawn from the same point in a perpendicular to a given line, cutting off unequal segments from the foot of the perpendicular, the more remote is the greater.

    SayProp. VIII is a "detour" theorem: a path that goes around is longer than one inside it.

  3. Draw the figure

    Prop. VIII: draw a segment AB and a point C above it; join CA and CB. Mark O inside the triangle and join OA and OB. Extend AO to meet CB at E.

    Prop. IX: draw a line AB, a perpendicular OC to it, and two lines OG and OE meeting AB, with CE longer than CG.

  4. Walk through the proof

    Prop. VIII: CA + CE > OA + OE, and BE + OE > OB (a straight line is shortest). Add the two inequalities, replace CE + BE by CB, and take OE from both sides.

    Prop. IX: mark CF = CG, so OF = OG. Extend OC to D with CD = OC and join ED and FD. Then OE = ED and OF = FD, and by Prop. VIII OE + ED > OF + FD. Halve both.

  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 § 102 together: only two equal straight lines can be drawn from a point to a straight line. Ask why there can't be three.

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