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Triangle Inequality Theorem

Rule explained

Can any 3 side lengths form a triangle?

question
NOPE!
As you can see in the picture below, it's not possible to create a triangle that has side lengths of 4 , 10 , and 3 .

triangle inequality theorem, impossible triangle
It turns out that there are some rules about the side lengths of triangles. You can's just make up 3 random numbers and have a triangle! You could end up with 3 lines like those pictured above that cannot be connected to form a triangle.

Video Tutorial on the Triangle Inequality Theorem

The Formula


Show Formula
The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side.
Triangle Inequality Theorem picture and formula

Note: This rule must be satisfied for all 3 conditions of the sides.
In other words, as soon as you know that the sum of 2 sides is less than (or equal to ) the measure of a third side, then you know that the sides do not make up a triangle .
triangle inequality theorem example 1
calc You can experiment for yourself using our free online triangle inequality theorem calculator -- which lets you enter any three sides and explains how the triangle inequality theorem applies to them.

Do I have to always check all 3 sets?

question
NOPE!
You only need to see if the two smaller sides are greater than the largest side!
Look at the eample above, the problem was that 4 + 3 (sum of smaller sides) is not greater than 10 (larger side)

Demonstrations illustrating the Triangle Inequality Theorem
The interactive demonstration below shows that the sum of the lengths of any 2 sides of a triangle must exceed the length of the third side. The demonstration also illustrates what happens when the sum of 1 pair of sides equals the length of the third side--you end up with a straight line! You can't make a triangle!


Otherwise, you cannot create a triangle from the 3 sides.

Demonstration 1: When the sum of 1 pair of sides exactly equals the measure of a 3rd side.

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Demonstration 2 When the sum of 1 pair of sides is less than the measure of a 3rd side.

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Practice Problems


Problem 1) Could a triangle have side lengths of
Side 1: 4
Side 2: 8
Side 3: 2
Answer


problem 2) Could a triangle have side lengths of
Side 1: 5
Side 2: 6
Side 3: 7
Answer


problem 3) Could a triangle have side lengths of
Side 1: 6
Side 2: 8
Side 3: 15
Answer
Load More Like These
Practice Problems II


problem 6) Two sides of a riangle have lengths 2 and 7. Find all possible lengths of the third side.
Answer


problem 7) Two sides of a riangle have lengths 12 and 5. Find all possible lengths of the third side.
Answer