Exercises
1. Make a conjecture about what other figures can be placed on the edges of a right triangle such that the sum of the areas of the figures on the legs equals the area of the figure on the hypotenuse? In deciding what figures work, use either Geometer's Sketchpad or Peanut Geometry to make your drawings.
2. What is the smallest number of matches needed to form simultaneously, on a plane, two different (non-congruent) Pythagorean triangles? The matches represent units of length and must not be broken or split in any way. (Gardner 160)
3. The following idea is taken from Algebra, A Challenging World, chapter 9. In the textbook, it asks students to think about the construction of a box kite and the lengths of the diagonal struts. However, students would enjoy actually building a kite and by constructing the kite, they work with measurements and the Pythagorean theorem.
The assignment: Design a box kite in the shape of a rectangular solid. This will require researching the kite-making process to determine what materials would best work for the kite. The opening at each end of the kite is a square. In making your kite a) determine the length of the diagonal struts using the Pythagorean theorem. This assignment is does not necessarily require the kites to fly, but instead allow students to be creative while learning math. b) If the square on the end of the solid has sides of length two then what is the length of the diagonal c) If the kite has squares with a side length of s, what is the length of the diagonal?
Angle A is 35 degrees. The sides adjacent to a are of length 3 and 5. What is the length of BC.
Solutions
1. Any figure can be drawn on the three sides as long as the three figures are similar. (Gardner 158)
2. (Gardner 162)
3. a) The length of the diagonal depends on the side length, so answers will vary.
b) If the side is of length 2 then the diagonal will be 2 times the square root of 2.
c) If the side length is s then the diagonal will be s times the square root of 2.
4. Using the law of cosines you get the formula a2 =32+52 –2(3)(5)cos35. Solve for an and one will get 3.07.