Triangle Inequality Theorem

Accueil » Triangle Inequality Theorem
Triangle Inequality Theorem: meaning, method and solved example (7th Grade) – Maths PDF

In short: The triangle inequality theorem says that any two sides of a triangle, added together, must beat the third side, which tells you at a glance whether three lengths can ever close up.

7th GradeGeometry7.G.A.2

What is Triangle Inequality Theorem?

The triangle inequality theorem explains why three sticks sometimes refuse to close into a triangle. When the two shorter sticks, laid end to end, still fall short of the longest one, nothing can join the loose ends.

For side lengths \(a\), \(b\) and \(c\), three statements must all hold: \(a + b > c\), \(a + c > b\) and \(b + c > a\). Usually one test is enough, the two shorter sides against the longest. The reason is simple: the straight path between two corners always beats a detour through the third corner.

How to spot it

  • A problem lists three lengths such as 5, 8 and 14 and asks whether a triangle with those sides exists.
  • Two sides are known and you must name the range of lengths the third side may take.
  • Sticks, straws, cables or fence sections of fixed sizes are to be joined into a closed loop.
  • One side looks suspiciously long next to the other two.

The method

  1. Line up the three lengths from smallest to largest.
  2. Add the two smaller lengths together.
  3. Compare that sum with the largest length; it has to be strictly bigger, otherwise no triangle exists.
  4. For an unknown third side beside known sides \(a\) and \(b\), it must land between \(|a – b|\) and \(a + b\), endpoints excluded.
Longest stick: 14 in5 in8 in5 + 8 = 13, and 13 is less than 14A 1-inch gap remains: no triangle.
Two sticks totaling 13 inches cannot span a 14-inch stick, so the ends never meet.

Triangle Inequality Theorem: solved example

Problem

The robotics club at an Ohio middle school owns aluminum rods cut to 5 inches, 8 inches and 14 inches. Will those three rods bolt together into a triangle?

Solution

The longest rod is 14 inches. The two others total \(5 + 8 = 13\).

Because 13 is less than 14, the pair falls one inch short and the free ends cannot meet. No triangle can be built. Trade the 14-inch rod for a 12-inch one and it works, since \(13 > 12\).

Try it

Question 1

Two fence sections of a triangular pen measure 9 feet and 15 feet. Between which two values must the third section lie?

Show the answer

It has to exceed \(15 – 9 = 6\) and stay under \(15 + 9 = 24\), so it lies strictly between 6 and 24 feet.

Question 2

Do lengths 7, 10 and 16 close into a triangle?

Show the answer

\(7 + 10 = 17\), and 17 beats 16, so yes; the two remaining comparisons hold as well.

Triangle Inequality Theorem vs. Pythagorean Theorem

The triangle inequality theorem holds for every triangle and only decides whether one can exist. The Pythagorean theorem needs a right angle and delivers an exact equation, \(a^2 + b^2 = c^2\). One adds plain lengths and uses a greater-than sign; the other squares the sides and uses an equals sign.

Pythagorean Theorem

Keep going

Lessons

Worksheets and quizzes

Related words

Whole dictionary: A–Z · 7th Grade · Geometry

Voter.. page