Wentworth's Plane and Solid Geometry · Book IV, Exs. 356–372 · Book 5: Book IV: Areas of Polygons
60. Original theorems on areas
Goal: Your student proves original theorems about areas of rectangles, parallelograms and trapezoids.
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)
- Squared (graph) paper
What's in the book
Pages 195–196: theorem exercises 356–372. The first three are the geometric forms of (a + b)², (a − b)² and a² − b², with their figures explained.
The lesson, step by step
Warm up
Ask: State Prop. X and its corollary.
Algebra in pictures
Ex. 356. The square constructed upon the sum of two straight lines is equivalent to the sum of the squares constructed upon these two lines, increased by twice the rectangle of these lines.
Draw the figure on squared paper with lines of 5 and 3 squares. Connect it to (a + b)² = a² + 2ab + b².
Two more
Do Ex. 357 and 358 the same way, matching each to its algebra.
On their own
Your student picks two of Ex. 359–372.
Ex. 359. The area of a rhombus is equal to half the product of its diagonals.
Wrap up
Your student explains Ex. 356 to you using the drawing.
Answers to the book's problems
Ex. 359: the diagonals cut the rhombus into two triangles on one diagonal as base, each with altitude half the other diagonal. Ex. 365: a diagonal cuts off a triangle that is half the parallelogram; the line joining the midpoints cuts from it a similar triangle with sides half as long, so one fourth of its area (§ 411), and one eighth of the parallelogram. Ex. 367: the line cuts the parallelogram into two parts made of pairs of equal triangles.
Worked out for this site from the scan. If a number in your copy differs, trust the book.