Exactly Reaching

Part of the Triangles section of Coddy's Geometry journey. Lesson 8 of 71.

What happens when the two shorter lengths add to exactly the longest, as 3, 4 and 7 do? The ends of the 3 and the 4 meet only when both lie flat along the 7. Then all three sides lie on one line, and there is no space inside them. No triangle is enclosed.

$Sticks of 3 and 4 lying flat along a stick of 7, meeting at a point on the line$

Sides 3 and 4 lying flat along a side of 7, with no triangle enclosed

So the test is strict. The two shorter sides must add to more than the longest, not equal to it. The lengths 3, 4 and 7 fail the test. The lengths 3, 4 and 6 pass. The lengths 3, 4 and 7.1 fail too. When you are given three lengths, sort them, add the two smallest, and check whether the sum is more than the largest. That one comparison tells you whether the lengths make a triangle.