Permutations

Question

A triangle can be classified according to its sides or according to its angles. There are three side length categories- equilateral, isosceles, and scalene- and three angle categories- acute, obtuse, and right.

     a. How many different relay teams can be formed?

     b. How many of these triangles are not possible? Which ones are they?

Answer 1

Not sure what "relay teams" means in the context of this question, but if you could freely select any of the 3 side categorizations and pair it up with any of the 3 angle categorizations there would be 3*3 = 9 formations  

 

If the triangle is equilateral then it is also equiangular which means all 3 angles are equal to 60 degrees so the triangle is acute.  So it is not possible to have an equilateral triangle which is an obtuse or right triangle.  

 

If the triangle is isosceles then 2 of the angles must be equal but they could be any 2 of the 3 angles.  So the triangle could be acute (80-50-50 degrees) or right (90-45-45 degrees) or obtuse (100-40-40 degrees).  

 

If the triangle is scalene then no 2 sides are equal which means that no 2 angles are equal.  Again the triangle could be acute (70-60-50 degrees) or right (90-60-30 degrees) or obtuse (110-40-30 degrees).  

 

So of the 9 formations 2 of them are not possible.