Wentworth's Plane and Solid Geometry · Book IV, Props. XV–XVII · Book 5: Book IV: Areas of Polygons
62. Squares equal to parallelograms; two lines from sum or difference and product
Goal: Your student can construct a square equivalent to a parallelogram, triangle or polygon, and a parallelogram equivalent to a square with a given sum or difference of base and altitude.
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 XV–XVII (§§ 421–425) with corollaries.
The lesson, step by step
Warm up
Ask: Construct the mean proportional between two lines.
The new idea
To construct a square equivalent to a given parallelogram.
To construct a parallelogram equivalent to a given square, and having the sum of its base and altitude equal to a given line.
To construct a parallelogram equivalent to a given square, and having the difference of its base and altitude equal to a given line.
Draw the figure
Prop. XV: lay off the altitude MN and the base NO in a line, draw a semicircle on MO, and erect NP. The square on NP is the answer.
Walk through the proof
XV: NP is the mean proportional between the altitude and base, so NP² = altitude × base (§ 370).
Corollaries: a triangle is turned into a square by using half its altitude; a polygon by first turning it into a triangle.
XVI: a line parallel to the diameter at the height of the square's side meets the semicircle where the perpendicular divides the diameter into the base and altitude.
XVII: a tangent equal to the side of the square, and the secant through the centre, give two lines differing by the diameter (§ 381).
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.
Wrap up
Read the notes: Prop. XVI finds two lines from their sum and product, Prop. XVII from their difference and product. Ask what equation from algebra this matches (a quadratic).