Question 234188
Given two sides with lengths 34 and 14, which of these lengths would NOT work for the third side of a triangle? 18, 34, 44, 22, 40.
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The sum of two sides of a triangle must always be greater than the third side.

So we'll make three checks on each choice.

Let's begin by trying 18.

Then the sides are 34, 14 and 18.

Add 34+14, that's 48, and that's greater than the third side, 18, so that
checks.

Add 34+18, that's 52, and that's greater than the third side, 14, so that
checks.

Add 14+18, that's 32, and that's NOT greater than the third side, 34, 
so that DOES NOT check.  So 18 will NOT work for the third side of
a triangle with sides 34 and 14.    

-----------------

Let's try 34.

Then the sides are 34, 14 and 34.

Add 34+14, that's 48, and that's greater than the third side, 34, so that
checks.

Add 34+34, that's 68, and that's greater than the third side, 14, so that
checks.

Add 14+34, that's 48, and that's greater than the third side, 34, so that
checks.

So 34 DOES work for a third side for a triangle with sides 34 and 14.

-------------------

Let's try 44.

Then the sides are 34, 14 and 44.

Add 34+14, that's 48, and that's greater than the third side, 44, so that
checks.

Add 34+44, that's 78, and that's greater than the third side, 14, so that
checks.

Add 14+44, that's 58, and that's greater than the third side, 34, so that
checks.

So 44 DOES work for a third side for a triangle with sides 34 and 14.

-------------------------

Let's try 22.

Then the sides are 34, 14 and 22.

Add 34+14, that's 48, and that's greater than the third side, 22, so that
checks.

Add 34+22, that's 56, and that's greater than the third side, 14, so that
checks.

Add 14+22, that's 36, and that's greater than the third side, 34, so that
checks.

So 22 DOES work for a third side for a triangle with sides 34 and 14.

-------------------

Let's try 40.

Then the sides are 34, 14 and 40.

Add 34+14, that's 48, and that's greater than the third side, 40, so that
checks.

Add 34+40, that's 74, and that's greater than the third side, 14, so that
checks.

Add 14+40, that's 54, and that's greater than the third side, 34, so that
checks.

So 40 DOES work for a third side for a triangle with sides 34 and 14.

---------------------

So the only choice that won't work for a third side of a triangle with
sides 34 and 14 is 18.

Why is that?

Suppose you had a triangle ABC drawn on the ground where AB is 34 feet,
AC is 14 feet, and BC is 18 feet.  Since the shortest distance from A
to B is the straight line segment connecting them, namely AB, or 34 feet.
But if there were such a triangle it would be shorter to walk from A to C,
and then turn and walk from C to B, since AC is 14 feet and BC is 18 feet,
so that would be only 32 feet.  So there can't be any such triangle.

Edwin</pre>