If x is an integer and 2<x<7, how many different triangles are there with sides of 2, 7, and x?

Question

The answer to this question is One. Why is that?

Answer 1

The two smaller sides must always add up to more than the largest side (otherwise you can't make a triangle). Since the sides must be *integer* sides, the only valid triangle is 2, 6 and 7.

Stated another way, here are the combinations:
2-3-7, impossible triangle (try connecting the 2 and 3 sides together...)
2-4-7, impossible triangle (try connecting the 2 and 4 sides together...)
2-5-7, straight line, not a triangle
2-6-7, valid triangle

So there is only one combination (2-6-7) that makes a triangle when x is an *integer* greater than 2 and less than 7.

Answer 2

Consider the triangle inequality theorem: The sum of any two sides of a triangle must be greater than the third. If you try x = 3 for ex., 3+7 > 2, but 2 + 3 < 7, so 3 is no good. Try the others and see what you find.

Answer 3

Because, the sum of a triangle's legs have to be larger than the hypotonuse. Since x need to be a leg, the only answer could be a 2 6 7 triangle

Answer 4

just one, for a triangle you need the sum of the smaller two sides to be greater than the largest side. only 6 would work for x since 5 + 2 is still only seven, and it needs to be LARGER than 7.