What is the pattern??

Question

Say I've been given two values for two sides of a triangle (9 and 12). What integer values of the 3rd side would make the triangle a right triangle, an obtuse triangle, and an acute triangle. What is the pattern to discovering this information?

Answer 1

Download a dynamic GeoGebra sketch to follow along here:
http://www.wyzant.com/resources/files/265695/explore_constraints_on_triangle_side_lengths

Triangle ABC has AB = 12 & BC = 9. Side AC can vary by rotating BC about B counterclockwise.

BC starts out lying on top of AB then when AC > 3 the triangle will be obtuse because angle C is obtuse.

Angle C becomes right when AC^2 + BC^2 = AB^2.

Then the triangle becomes acute until angle B is right when AB^2 + BC^2 = AC^2.

Then the triangle becomes obtuse until BC is in opposite direction to AB but collinear, then there is no triangle.

Answer 2

It makes right triangle if

 

  3rd side equals  :√ (X ^2 + Y^2) = √Z^2=  Z

                           

                            √(9^2 + 12^2) = 15

 

      Z^2 = X ^2 + Y^2 - 2 X. Y Cos Θ / Θ is the angle between X, Y

 

       If angle between X , Y is obtuse , then Cos Θ <0 and Z > 15

 

                        If  Θ is acute then Cosθ >0 and  Z<15