Wentworth's Plane and Solid Geometry · Book IV, Props. IV–VII · Book 5: Book IV: Areas of Polygons
58. Parallelograms, triangles and trapezoids
Goal: Your student can prove the area formulas for the parallelogram, triangle and trapezoid, and compare triangles with an equal angle.
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
- Scissors
What's in the book
Propositions IV–VII (§§ 400–410) with corollaries, and Exercises 353–354.
The lesson, step by step
Warm up
Ask: What is the area of a rectangle, and why?
The new idea
The area of a parallelogram is equal to the product of its base by its altitude.
The area of a triangle is equal to half the product of its base by its altitude.
The area of a trapezoid is equal to half the sum of its bases multiplied by the altitude.
The areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles.
Draw the figure
Draw a slanting parallelogram on squared paper. Cut a right triangle off one end and slide it to the other end to make a rectangle. Then draw a diagonal of a parallelogram to make two triangles.
Walk through the proof
IV: the two right triangles at the ends are equal, so taking one or the other from the same figure leaves the rectangle or the parallelogram: they are equivalent.
V: a triangle is half of a parallelogram on the same base and altitude (§ 179).
VI: a diagonal cuts the trapezoid into two triangles with the same altitude.
VII: join B to E; compare the triangles two at a time, each pair having an equal altitude.
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.
Exercises
Ex. 353. The lines which join the middle point of either diagonal of a quadrilateral to the opposite vertices divide the quadrilateral into two equivalent parts.
Wrap up
Read § 409: any irregular polygon can be measured by cutting it into triangles.
Answers to the book's problems
Ex. 353: each half of the quadrilateral is made of two triangles whose bases are the equal halves of the diagonal and whose altitudes match (§ 404).
Worked out for this site from the scan. If a number in your copy differs, trust the book.