Triangle Inequality Math Example 2

Follow the full solution, then compare it with the other examples linked below.

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Two sides of a triangle have lengths 8 and 13. Find the range of possible lengths for the third side.

1
Step 1: Let the third side have length $x$ . Apply the Triangle Inequality to all three combinations.
2
Step 2: $8 + 13 > x \Rightarrow x < 21$ .
3
Step 3: $8 + x > 13 \Rightarrow x > 5$ .
4
Step 4: $13 + x > 8$ is always true for positive $x$ .
5
Step 5: Combine: $5 < x < 21$ .

Answer

The third side must satisfy

5 < x < 21

When two sides of a triangle are known, the third side must be strictly between their difference and their sum:

|a - b| < c < a + b

. Here,

|13 - 8| = 5

and

13 + 8 = 21

, so

5 < x < 21

. The endpoints are excluded because equality would make the 'triangle' degenerate (a straight line).

About Triangle Inequality

The sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side.

Can sides of length 4, 7, and 10 form a triangle? Check all three triangle inequality conditions.

Which of these sets of side lengths cannot form a triangle? (a) 3, 4, 5. (b) 1, 2, 3. (c) 5, 8, 12.

In [formula], [formula] and [formula]. The perimeter is 24. Is this a valid triangle? Find [formula]