A box contains one 2-inch rod, one 3-inch rod, one 4-inch rod, and one 5-inch rod. What is the maximum number of different triangles that can be made using these rods as sides?
In order for a, b, and c to be the sides of a triangle,
all three of these must be true:
a + b > c, a + c > b, and b + c > a
That is, in any triangle, the sum of any two sides must
be greater than the third side.
We will make all combinations of three of the rods
and eliminate any that can't be triangles:
1. 2-inch, 3 inch, 4-inch
That is a triangle, because 2+3 > 4, 2+4 > 3, and 3+4 > 2
2. 2-inch, 3 inch, 5-inch
This is NOT a triangle because even though 2+5>3, and 3+5>2,
however 2+3 is NOT GREATER than 5, for it's EQUAL to 5. And
unless all three are true, they cannot form a triangle.
3. 2-inch, 4 inch, 5-inch
That is a triangle, because 2+4 > 5, 2+5 > 4, and 4+5 > 2
4. 3-inch, 4 inch, 5-inch
That is a triangle, because 3+4 > 5, 3+5 > 4, and 4+5 > 3
So a maximum of 3 triangles can be formed with those
rods, since we eliminated case 2 above.
Edwin