Question 578489: Which of the following cannot be the lengths of the sides of a triangle?
1) 2,5,6
2)3,4,5
3)4,5,6,
4)4,5,10
How can I figure out the answer of this question if it is not referring to right triangles?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! Add up the lengths of the two shortest sides. If it is not more than the length of the longest side, there is no triangle.
The two shortest sides (like any two sides) of a triangle are joined en to end at one of the vertices of the triangle. A segment connecting the other two "free" ends would be the third side of the triangle. As you vary the angle the joined first two sides form, the farthest that the free ends will be from each other is the sum of the lengths. And in that case, they will be forming a  angle, which does not make a triangle.
Looking at it another way, the shortest path between two points is a straight line. SO the shortest path from one vertex of a triangle to another is the side connecting those two vertices. Going down the other two sides, that path would be the sum of those two sides' length and it is longer than the direct straight way.
So:
1) 2+5=7>6 triangle can be formed
2) 3+4=7>5 triangle can be formed and it is a right triangle
3) 4+5=9>6 triangle can be formed
4) 4+5=9<10 triangle cannot be formed
 The segments of length 4 and 5 can swing circles around the ends of the segment of length 10, but cannot meet and close the triangle.
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