Wentworth's Plane and Solid Geometry · Book IV, Exs. 387–403 · Book 5: Book IV: Areas of Polygons
64. Transforming figures
Goal: Your student transforms triangles, parallelograms and squares into equivalent figures of a required shape, and divides triangles into equal parts.
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
Page 207: construction exercises 387–403, beginning with a worked one (Ex. 387).
The lesson, step by step
Warm up
Ask: Why are triangles with the same base and the same altitude equivalent?
The model
Work through Ex. 387 and its printed hint together:
Ex. 387. To transform a triangle ABC into an equivalent triangle, having a side equal to a given length l, and an angle equal to angle BAC.
On their own
Your student picks two or three: Ex. 391, 394, 398, 400 or 402.
Ex. 400. To divide a given triangle into two equivalent parts by a line through a given point P in one of the sides.
Check
For a division into equal parts, measure bases and altitudes afterwards and compute both areas.
Wrap up
Your student explains one construction aloud.
Answers to the book's problems
Ex. 391: turn the triangle into a square (§ 422), then use Prop. XVIII with any equilateral triangle as P. Ex. 394: the new base is the fourth proportional to the given altitude, the parallelogram's altitude and its base. Ex. 402: the cutting line must cut off a similar triangle of half the area, so its side is the side of the triangle divided by √2 (Prop. XX).
Worked out for this site from the scan. If a number in your copy differs, trust the book.