Triangle Inequality Theorem

The sum any two sides in a triangle is always greater than the third side of triangle.

If one side is longer than the sum of other two sides as shown below, then it is not a triangle.

See the example below:

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In the above case, AD + BC = 18 + 9 = 27 units, which is less than side AB of length 30 units. Here since one side AB is longer, so the other two sides, AD and BC don’t meet.

In the above triangle, following holds true:

b + c > a

a + b > c

a + c > b

Example: What is the possible range for x?

According to Triangle Inequality Theorem, the sum any two sides in a triangle is always greater than the third side of triangle.

So, the three inequalities for the above triangle are:

(7 – x) + 7 > 3x => 14 – x > 3x => 14 > 4x => $\frac{7}{2}$ x =>x < $\frac{7}{2}$
3x + (7 – x) > 7 => 2x + 7 > 7 => 2x > 0 => x > 0
3x + 7 > 7 – x => 4x > 0 => x > 0

Combining the three inequalities, we get the possible range as 0 < x < $\frac{7}{2}$

Check Point

Can the numbers 4, 5, 10 be the sides of a triangle?
If 13 and 8 are the two sides of a triangle, then what is the possible range for the third side?
Do the side lengths, 8, 11, and 16 form a triangle?
If 3 + 2x, 7 – x and 2x are the sides of a triangle, then what is the possible range for x?
If 10 and 15 are the two sides of a triangle, then what is the possible range for the third side?

Answer key

No
5 < x < 21
Yes
$\frac{4}{5}$ < x < 10
5 < x < 25

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