If two sides of a triangle are 6 and 16, what is the range of the possible lengths of the third side?

10 ≥ x ≥ 22

Step-by-step explanation:

Use the triangle inequality theorem to state the range of possibilities for the third side.  If the sides of a triangle are a, b, and c, then the theorem states:

a + b ≥ c,

b + c ≥ a,

a + c ≥ b

Let's put it together then:

6 + 16 ≥ x,

so 22 ≥ x.

x + 6  ≥ 16, so

x ≥ 10.

16 + x ≥ 6, so

x ≥ -10

The 2 things in math that will never EVER be negative are times and distances, so the -10 can be disregarded.

That gives you the range of 10 ≥ x ≥ 22

That means that the third side cannot be shorter than 10 or else the triangle will not close; it also means that the third side cannot be longer than 22, or it will close but leave an "overhang".  

Keep in mind that this is NOT for right triangles!  To see what the third side of a right triangle is you would use Pythagorean's Theorem.

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