A demonstration of when you can and can not use the Pythagorean theorem, its converse, and the Pythagorean inequalities theorem.[br][br]Warning: In the problems below, make sure to distinguish between [math]a[/math], [math]b[/math], and [math]c[/math] and [math]a^{2}[/math], [math]b^{2}[/math], and [math]c^{2}[/math].

1. Demonstration of the converse of the Pythagorean theorem. (Also note that if <C is a right angle, the Pythagorean theorem tells you [math]a^{2}+b^{2}=c^{2}[/math]).[br](a) Change a, b, and c such that [math]a^{2}+b^{2}=c^{2}[/math] (as in the default example).[br](b) What do you notice about <C and [math]\triangle[/math]ABC?[br][br]2. Demonstration of the Pythagorean inequalities theorem (Part 1 - Theorem 8-5).[br](a) Change a, b, and c such that [math]a^{2}+b^{2}<c^{2}[/math].[br](b) What do you notice about <C and [math]\triangle[/math]ABC?[br](c) Aside from proving this theorem, one way to remember it is to notice that [math]a^{2}+b^{2}<c^{2}[/math] roughly means that c is [b][i]sufficiently larger[/i][/b] than a and b. When this situation happens, what kind of triangle comes into your mind?[br][br]3. Demonstration of the Pythagorean inequalities theorem (Part 2 - Theorem 8-4).[br](a) Change a, b, and c such that [math]a^{2}+b^{2}>c^{2}[/math].[br](b) What do you notice about <C and [math]\triangle[/math]ABC?[br](c) Aside from proving this theorem, one way to remember it is to notice that [math]a^{2}+b^{2}>c^{2}[/math] roughly means that c is [b][i]sufficiently smaller[/i][/b] than a and b. When this situation happens, what kind of triangle comes into your mind?[br][br]4. Demonstration of an [b][i]improper[/i][/b] way of using the converse of the Pythagorean theorem and the Pythagorean inequalities theorem.[br](a) Let [math]a^{2}=16[/math], [math]b^{2}=32[/math], and [math]c^{2}=16[/math]. What kind of triangle is [math]\triangle[/math]ABC?[br](b) Let [math]a^{2}=16[/math], [math]b^{2}>32[/math], and [math]c^{2}=16[/math]. What kind of triangle is [math]\triangle[/math]ABC?[br](c) When c<a or c<b, why can you [i][b]not[/b][/i] compare [math]a^{2}+b^{2}[/math] with [math]c^{2}[/math] to use the converse of the Pythagorean theorem or the Pythagorean inequalities theorem?[br][br]5. Demonstration of impossible triangle side lengths.[br](a) Change a, b, and c such that one of them is nonpositive (0 or less than 0). Why is this impossible?[br](b) Change a, b, and c such that a+b<c, a+c<b, or b+c<a. Why is this impossible?