Twelve points are spaced evenly around a circle, lettered from a to l. let n be the total number of isosceles triangles, including equilateral triangles, that can be constructed from three of these points. a different orientation of the same lengths counts as a different triangle, because a different combination of points form the vertices. what is the value of n?

  N = 52

Step-by-step explanation:

We assume that a 1-5-9 equilateral triangle is indistinguishable from a 5-9-1 equilateral triangle, so only count it once.

There are 5 isosceles triangles that can be created using each lettered vertex as the apex. Of those, 1 is equilateral, so there are 4×12 non-equilateral isosceles triangles that can be formed.

There are 4 unique equilateral triangles, so the total number is ...

  N = 4×12 + 4 = 52

the sum of the lengths of any two sides of a triangle is greater than the length of the third side. if you take the three sides of a triangle and add them in pairs, the sum is greater than (not equal to) the third side. if that is not true, then it is not possible to construct a triangle with the given side lengths.

step-by-step explanation: