Practice Triangle Inequality in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

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

Try to build a triangle with two short sticks and one very long one—you can't. The two short sticks can't reach across to close the shape. It's like trying to take a shortcut: the direct path (one side) is always shorter than going around (the other two sides combined).

Showing a random 20 of 50 problems.

Example 1

medium
Two sides of a triangle measure 99 and 1414. List all integer values the third side could take.

Example 2

hard
The perimeter of a triangle is 3030. Two sides are 1111 and 1212. Is the triangle valid? What is the third side?

Example 3

hard
In △ABC\triangle ABC, AB=6AB = 6 and BC=10BC = 10. The perimeter is 24. Is this a valid triangle? Find ACAC and verify.

Example 4

medium
A person walks 8 km east then 5 km in some direction. What is the range of possible straight-line distances from start?

Example 5

medium
A triangle has sides 9, 12, and x where x is an integer. How many integer values of x are possible?

Example 6

easy
Two sides of a triangle are 22 and 99. Which of 77, 88, 1111 could be the third side?

Example 7

easy
Can sides 11, 11, and 33 form a triangle?

Example 8

medium
A triangle has sides aa, a+3a + 3, and a+5a + 5. Find the smallest integer value of aa for which the triangle exists.

Example 9

hard
A triangle has integer side lengths and perimeter 1515. If one side is 66, find the number of possible triangles.

Example 10

challenge
A pentagon has sides that are all positive. Explain why the longest side must be less than the sum of the other four sides (a polygon inequality).

Example 11

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

Example 12

hard
Three positive real numbers a,b,ca, b, c satisfy a2+b2+c2=50a^2 + b^2 + c^2 = 50 and a+b+c=12a + b + c = 12. Could they form a triangle?

Example 13

easy
Can sides of length 3, 4, and 5 form a triangle?

Example 14

easy
Two sides of a triangle are 1.51.5 and 2.52.5. What is the range for the third side?

Example 15

medium
If two sides of a triangle are aa and bb with a≥ba \geq b, what is the range for the third side cc?

Example 16

easy
Two sides of a triangle are 7 and 10. What is the range of possible lengths for the third side?

Example 17

easy
A triangle has two sides of length 55 each. What is the range for the third side?

Example 18

medium
A triangle has sides 77, 2424, xx and is a right triangle. Use the triangle inequality and Pythagoras to find xx.

Example 19

easy
What does the triangle inequality state about the three sides of a triangle?

Example 20

easy
Two sides of a triangle measure 66 and 66. Find the maximum integer value for the third side.
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