As you can see in the picture below, it's not possible to create a triangle that has side lengths of 4, 8, and 3
It turns out that there are some rules about the side lengths of triangles. You can't 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.
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.
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.
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.
You only need to see if the two smaller sides are greater than the largest side!
Look at the example above, the problem was that 4 + 3 (sum of smaller sides) is not greater than 10 (larger side)
We start using this shortcut with practice problem 2 below.
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.
Use the triangle inequality theorem and examine all 3 combinations of the sides. As soon as the sum of any 2 sides is less than the third side then the triangle's sides do not satisfy the theorem.
Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.
| small + small > large |
because 5 + 6 > 7
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Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.
| small + small > large |
because 1.2 + 1.6 $$\color{Red}{ \ngtr } $$ 3.1
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Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.
| small + small > large |
because 6 + 8 $$\color{Red}{ \ngtr } $$ 16
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More like Problem 1-4...
Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.
| small + small > large |
because 5 + 5 $$\color{Red}{ \ngtr } $$ 10
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Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.
| small + small > large |
because 7 + 9 > 15
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Two sides of a triangle have lengths 8 and 4. Find all possible lengths of the third side.
difference $$< x <$$ sum
$$8 -4 < x < 8+4 $$
There's an infinite number of possible triangles, but we know that the side must be larger than 4 and smaller than 12 .
difference $$< x <$$ sum
$$7 -2 < x < 7+2$$
difference $$< x <$$ sum
$$12 -5 < x < 12 + 5$$
