Key takeaway: The triangle inequality theorem says that no side of a triangle can be as long as the other two sides put together, since two shorter sides must outreach the third.
Grade 77.G.A.2Geometry and Measurement
Definition of Triangle Inequality Theorem
The triangle inequality theorem captures a simple fact about travel: the straight route between two towns is never longer than a detour through a third town. Walk along two edges of a triangular park and you cover more ground than walking the remaining edge.
So three lengths build a triangle only when the two shorter ones, added up, beat the longest. If the total only ties the longest, everything collapses onto one line. If the total falls short, the two shorter pieces dangle without touching.
Formula and notation
Name the sides \(a\), \(b\) and \(c\). A triangle exists only when each of these is true:
\[a + b > c,\qquad a + c > b,\qquad b + c > a\]
Shortcut: add the two smallest lengths and see whether the sum beats the biggest one.
Key facts
- Comparing the two shortest sides with the longest settles the question alone.
- A tie, where the shorter sides sum exactly to the longest, produces a flat segment.
- A missing side must be greater than the difference and less than the sum of the other two.
- The rule applies to every triangle: acute, right, obtuse, scalene or isosceles.
Triangle Inequality Theorem example
Park rangers have trail segments of 7 miles, 12 miles and 18 miles. Does the triangle inequality theorem allow a triangular loop? A second loop uses trails of 8 and 13 miles; what lengths can its third trail have?
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Add the two shortest: \(7 + 12 = 19\). Since \(19 > 18\), the loop works.
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For 8 and 13, the third side \(x\) must exceed the difference: \(x > 13 – 8 = 5\).
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It must also be less than the sum: \(x < 13 + 8 = 21\).
The 7-12-18 loop is possible, and the third trail must satisfy \(5 < x < 21\).
Common mistake
Watch out
Hana tests sides 6, 7 and 20 by checking \(6 + 20 > 7\) and \(7 + 20 > 6\), then says a triangle exists.
Why it is wrong
She skipped the test that fails. The one that matters involves the longest side: \(6 + 7 = 13\), which is less than 20. All three inequalities must hold, so no triangle can be built from 6, 7 and 20.
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Geometry and Measurement