Triangle Inequality: Step-by-Step Guide
Not every set of three lengths can make a triangle. The triangle inequality says that any two sides added together must be longer than the third side. If you know two sides, a and b, the third side x must be bigger than the difference between a and b, and smaller than their sum: |a - b| < x < a + b.
Math Racer skill guide · Geometry · Grade 10
How to solve it, step by step
- Write down the three side lengths. Call them a, b, and c.\[a,\; b,\; c\]
- Check all three pairs. Add each pair of sides and make sure the sum is bigger than the third side.\[a + b > c,\quad a + c > b,\quad b + c > a\]
- Find the range for a missing side. If you know two sides, a and b, the third side x must be bigger than |a - b| and smaller than a + b.\[|a - b| < x < a + b\]
- Decide if a triangle is possible. If all three checks are true, the sides make a triangle. If a sum is equal to the third side, the sides lie flat in a line. If a sum is smaller, the sides cannot reach each other.\[\text{valid} \iff a + b > c \;\land\; a + c > b \;\land\; b + c > a\]
Worked example
Two sides of a triangle measure 7 and 10. Find the range of possible lengths for the third side.
Step 1: The two known sides are 7 and 10. Call the third side x.
Step 2: The rule is |a - b| < x < a + b.
Step 3: The difference is |7 - 10| = 3.
Step 4: The sum is 7 + 10 = 17.
Step 5: So the third side must be between 3 and 17: 3 < x < 17.
Check: Try x = 10, which is in the range. 7 + 10 > 10, 7 + 10 > 10, and 10 + 10 > 7. All true, so it makes a triangle.
The third side must be bigger than 3 and smaller than 17.
Tips and common mistakes
- Check all three pairs. Checking only one is not enough.
- If two sides add up to exactly the third, the sides lie in a straight line and make no real triangle.
- The longest side is across from the biggest angle. The shortest side is across from the smallest angle.
- The third side is always strictly between |a - b| and a + b. The ends are not included.
- If the sides must be whole numbers, the choices run from |a - b| + 1 up to a + b - 1.
- This rule works for every triangle, not just right triangles.
- Quick check: the longest side must be shorter than the other two added together.
- Do not mix this up with the Pythagorean theorem. The triangle inequality works for all triangles. The Pythagorean theorem is only for right triangles.