Triangles

vladmire wrote:The sum of the 2 shortest sides of any triangle cannot be greater than the longest side correct? This is how I came up with none for this problem correct?

If two sides of a triangle have lengths 2 and 5, which of the following could be the perimeter of the triangle?

a.9
b.15
c.19
none.

lunarpower wrote:

hwiya320 wrote: sum of any 2 sides are GREATER than 3rd side.

yep.

this is actually an offshoot of something you probably learned sometime around first grade:
the shortest path from one point to another is a straight line (segment).

if you draw a triangle, then notice that each side of the triangle is a straight path from one vertex to another.
the other 2 sides, taken together, are a non-straight path between the same two vertices.
therefore, the other 2 sides are longer than the chosen side.
that is all.

--

btw, there's another, equally useful, inequality that you should also know about triangles:

THIRD SIDE INEQUALITY
if you KNOW two sides of a triangle, then
DIFFERENCE < third side < SUM

this can be derived directly from the triangle inequality (the thing discussed above), but it's worth memorizing separately; the last thing you want to do on a 2-minute problem is start deriving new rules.

in this problem, this rule immediately yields
5 - 2 < third side < 5 + 2
3 < third side < 7
10 < perimeter < 14 (add 7 to both sides)
done.