8.1 Pythagorean Thm & Converse & Pythagorean Ineq Thm
- Author:
- soongy
- Topic:
- Inequalities
A demonstration of when you can and can not use the Pythagorean theorem, its converse, and the Pythagorean inequalities theorem.
Warning: In the problems below, make sure to distinguish between , , and and , , and .
1. Demonstration of the converse of the Pythagorean theorem. (Also note that if <C is a right angle, the Pythagorean theorem tells you ).
(a) Change a, b, and c such that (as in the default example).
(b) What do you notice about <C and ABC?
2. Demonstration of the Pythagorean inequalities theorem (Part 1 - Theorem 8-5).
(a) Change a, b, and c such that .
(b) What do you notice about <C and ABC?
(c) Aside from proving this theorem, one way to remember it is to notice that roughly means that c is sufficiently larger than a and b. When this situation happens, what kind of triangle comes into your mind?
3. Demonstration of the Pythagorean inequalities theorem (Part 2 - Theorem 8-4).
(a) Change a, b, and c such that .
(b) What do you notice about <C and ABC?
(c) Aside from proving this theorem, one way to remember it is to notice that roughly means that c is sufficiently smaller than a and b. When this situation happens, what kind of triangle comes into your mind?
4. Demonstration of an improper way of using the converse of the Pythagorean theorem and the Pythagorean inequalities theorem.
(a) Let , , and . What kind of triangle is ABC?
(b) Let , , and . What kind of triangle is ABC?
(c) When c<a or c<b, why can you not compare with to use the converse of the Pythagorean theorem or the Pythagorean inequalities theorem?
5. Demonstration of impossible triangle side lengths.
(a) Change a, b, and c such that one of them is nonpositive (0 or less than 0). Why is this impossible?
(b) Change a, b, and c such that a+b<c, a+c<b, or b+c<a. Why is this impossible?