Wentworth's Plane and Solid Geometry · Book IV, Props. XI–XIV · Book 5: Book IV: Areas of Polygons
61. Constructing squares and triangles of given area
Goal: Your student can construct squares equal to the sum and difference of two squares, a similar polygon equal to the sum of two similar polygons, and a triangle equal to a polygon.
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 XI–XIV (§§ 417–420), and Exercises 373–380.
The lesson, step by step
Warm up
Ask: State the theorem of the square on the hypotenuse.
The new idea
To construct a square equivalent to the sum of two given squares.
To construct a square equivalent to the difference of two given squares.
To construct a triangle equivalent to a given polygon.
Prop. XIII (§ 419): to construct a polygon similar to two given similar polygons and equivalent to their sum.
Draw the figure
Prop. XI: draw a right angle; mark legs of 3 cm and 4 cm; the hypotenuse is the side of the required square.
Prop. XIV: draw an irregular pentagon. Cut off one vertex at a time by the book's method: a diagonal, a parallel to it through the cut-off vertex, and a new side.
Walk through the proof
XI and XII come straight from Prop. X and its corollary.
XIII: similar polygons are as the squares of their sides, so build a right triangle on the two homologous sides.
XIV: triangles on the same base with vertices on a parallel to the base are equivalent (§ 404). Each step removes one side without changing the area.
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.
Exercises
Ex. 373. If the perimeter of a rectangle is 72 feet, and the length is equal to twice the width, find the area.
Ex. 380. A rhombus contains 100 square feet, and the length of one diagonal is 10 feet. Find the length of the other diagonal.
Wrap up
Ask: Why can we slide the vertex of a triangle along a parallel to its base without changing its area?
Answers to the book's problems
Ex. 373: 288 sq ft. Ex. 374: 11,008 tiles. Ex. 375: 39√3, about 67.55 sq ft. Ex. 376: a square with side 5 in. Ex. 377: a square with side 1½ in. Ex. 378: 40 ft. Ex. 379: 32 ft. Ex. 380: 20 ft.
Worked out for this site from the scan. If a number in your copy differs, trust the book.