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 9 and 14. List all integer values the third side could take.
Example 2
hard
The perimeter of a triangle is 30. Two sides are 11 and 12. Is the triangle valid? What is the third side?Perimeter is 30; find the third side x and verify the triangle is valid.
Example 3
hard
In △ABC, AB=6 and BC=10. The perimeter is 24. Is this a valid triangle? Find AC 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?How many integer values of x satisfy the triangle inequality?
Example 6
easy
Two sides of a triangle are 2 and 9. Which of 7, 8, 11 could be the third side?
Example 7
easy
Can sides 1, 1, and 3 form a triangle?
Example 8
medium
A triangle has sides a, a+3, and a+5. Find the smallest integer value of a for which the triangle exists.
Example 9
hard
A triangle has integer side lengths and perimeter 15. If one side is 6, 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,c satisfy a2+b2+c2=50 and a+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 1.5 and 2.5. What is the range for the third side?Find the range 1 < x < 4 for the third side.
Example 15
medium
If two sides of a triangle are a and b with a≥b, what is the range for the third side c?
Example 16
easy
Two sides of a triangle are 7 and 10. What is the range of possible lengths for the third side?Find the range of possible lengths for the third side x.
Example 17
easy
A triangle has two sides of length 5 each. What is the range for the third side?Find the range of valid lengths for the third side x.
Example 18
medium
A triangle has sides 7, 24, x and is a right triangle. Use the triangle inequality and Pythagoras to find x.
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 6 and 6. Find the maximum integer value for the third side.Find the maximum integer value for the third side x.