You can know if side lengths make a triangle by using the triangle inequality theorem: the sum of any two side lengths must be greater than the third side. A quick shortcut is to compare the longest side with the sum of the other two sides.
Introduction
A triangle is a closed, three-sided polygon. Not every group of three numbers can form one. As an example, lengths of 2, 3, and 10 cannot create a triangle because the two shorter sides cannot reach each other when placed at opposite ends of the longest side That's the whole idea..
The rule used to test this is called the triangle inequality theorem. It states that for a valid triangle, the sum of every pair of side lengths must be greater than the remaining side. If this condition is true for all three pairs, the lengths can form a triangle That alone is useful..
The Triangle Inequality Theorem
Suppose the three side lengths are represented by a, b, and c. They form a triangle only when all three statements are true:
- a + b > c
- a + c > b
- b + c > a
Each side must be shorter than the combined length of the other two sides. This prevents a side from being so long that the remaining sides cannot meet Worth knowing..
A Faster Method
You do not always need to test all three inequalities. First, identify the longest side. Then compare it with the sum of the two shorter sides:
- If the longest side is less than the sum of the other two sides, the lengths form a triangle.
- If the longest side is equal to the sum of the other two sides, they form a straight line, not a triangle.
- If the longest side is greater than the sum of the other two sides, they cannot form a triangle.
This shortcut works because if the longest side is shorter than the combined length of the other two sides, the other two inequalities will automatically be true Which is the point..
Steps to Determine Whether Three Lengths Form a Triangle
1. Confirm That Every Length Is Positive
A side length must be greater than zero. A length of zero would produce no side, while a negative length has no physical meaning in geometry It's one of those things that adds up..
Here's one way to look at it: 4, 0, and 5 cannot form a triangle because one length is zero And that's really what it comes down to..
2. Find the Longest Side
Arrange the lengths from smallest to largest. This makes it easier to identify the side that must be tested.
For 8, 5, and 12, the ordered lengths are:
- 5
- 8
- 12
The longest side is 12.
3. Add the Two Shorter Sides
Add the lengths of the two smaller sides:
5 + 8 = 13
4. Compare the Sum With the Longest Side
Now compare 13 with the longest side, 12:
13 > 12
Because the sum of the shorter sides is greater than the longest side, 8, 5, and 12 can form a triangle.
Worked Examples
Example 1: Lengths That Form a Triangle
Test 6, 7, and 9.
The longest side is 9. Add the two shorter sides:
6 + 7 = 13
Because 13 > 9, these lengths satisfy the triangle inequality theorem. That's why, 6, 7, and 9 form a triangle.
To verify all three inequalities:
- 6 + 7 > 9
- 6 + 9 > 7
- 7 + 9 > 6
All three statements are true Simple as that..
Example 2: Lengths That Do Not Form a Triangle
Test 4, 5, and 10.
The longest side is 10. Add the two shorter sides:
4 + 5 = 9
Because 9 < 10, the shorter sides cannot meet. Because of this, 4, 5, and 10 do not form a triangle.
One of the required inequalities fails:
4 + 5 > 10
Since 9 > 10 is false, the three lengths are invalid Less friction, more output..
Example 3: Equal Sums Create a Degenerate Case
Test 3, 6, and 9.
Add the two shorter sides:
3 + 6 = 9
The sum is equal to the longest side. Day to day, if these segments are placed end to end, they create one straight segment of length 9. There is no enclosed space and no interior angle greater than 0°.
That's why, 3, 6, and 9 do not form a normal triangle. They form what is sometimes called a degenerate triangle.
Example 4: Repeated Side Lengths
Test 5, 5, and 5.
The longest side is 5, and the sum of the other two sides is:
5 + 5 = 10
Because 10 > 5, the lengths form a triangle. In fact, they form an equilateral triangle, which has three equal sides.
Repeated lengths are allowed. An isosceles triangle may have two equal sides, and an equilateral triangle has three equal sides It's one of those things that adds up..
Why the Triangle Inequality Theorem Works
Imagine placing the longest side on a flat surface. On top of that, the two shorter sides must be attached to its endpoints and meet above or below it. If their combined length is too small, there will be a gap between their free ends Most people skip this — try not to..
If their combined length is exactly equal to the longest side, the shorter sides lie directly along it. Here's the thing — they meet, but only in a straight line. This produces no triangular region.
Only when their combined length is greater than the longest side can the two sides meet at an angle. That angle creates the enclosed space required for a triangle Most people skip this — try not to. Still holds up..
This relationship also explains why one side of a triangle can never be as long as—or longer than—the other two sides combined.
Testing Fractions and Decimals
The same method works with fractions and decimals.
Decimal Example
Test 2.4, 3.1, and 5.6 It's one of those things that adds up..
The longest side is 5.6. Add the shorter sides:
2.4 + 3.1 = 5.5
Because 5.5 < 5.6, these lengths do not form a triangle.
Fraction Example
Test 1/2, 2/3, and 4/5.
The longest side is 4/5. Add the shorter sides using a common denominator:
1/2 + 2/3 = 3/6 + 4/6 = 7/6
Because 7/6 > 4/5, the lengths can form a triangle Simple, but easy to overlook. No workaround needed..
When comparing fractions, use equivalent fractions, decimals, or cross-multiplication to make the comparison accurate That's the part that actually makes a difference..
Finding a Possible Range for a Missing Side
The triangle inequality
