Lesson 4 · The Triangle

Section 03

Triangle Inequality

Not every set of three lengths can be bent into a triangle. Two short sides must be able to reach across the longest one.

Rule

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
6 cm4 cm5 cmABC
In triangle ABC: AB + BC > AC, AB + AC > BC, AC + BC > AB

The fast test

You do not need to check all three inequalities. Add the two shortest sides and compare with the longest:

  • short + short > long → the triangle can be drawn.
  • short + short ≤ long → the triangle cannot be drawn.

Activity: draw a triangle from three lengths

A compass is a drawing tool for circles, and it is also used to transfer exact lengths. Try each set on paper.

6 cm, 5 cm, 4 cm — it works

  1. 1. Use a ruler to draw AB = 6 cm.
  2. 2. Set the compass to 5 cm, put the needle at B and draw an arc.
  3. 3. Set the compass to 4 cm, put the needle at A and draw a second arc; the arcs meet at C.
  4. 4. Join AC and BC. The triangle exists because 4 + 5 = 9 > 6.

6 cm, 3 cm, 2 cm — impossible

  1. 1. Draw AB = 6 cm.
  2. 2. Arc of 3 cm from B, arc of 2 cm from A.
  3. 3. The arcs never meet — they fall short of each other.
  4. 4. No triangle, because 2 + 3 = 5 < 6.

6 cm, 3 cm, 3 cm — still impossible

  1. 1. Draw AB = 6 cm.
  2. 2. Arc of 3 cm from B, arc of 3 cm from A.
  3. 3. The arcs touch exactly at the midpoint of AB — the three points are collinear.
  4. 4. No triangle, because 3 + 3 = 6 (equal, not greater).

Example 4 — Can these be side lengths?

  1. 3 cm, 4 cm, 5 cm → 3 + 4 = 7 > 5 ✔ triangle can be drawn
  2. 4 cm, 6 cm, 11 cm → 4 + 6 = 10 < 11 ✘ triangle cannot be drawn
  3. 7 cm, 9 cm, 14 cm → 7 + 9 = 16 > 14 ✔ triangle can be drawn

The third side always lives in a range

The length of any side is greater than the difference of the other two and smaller than their sum.

Example 5 — Largest possible third side

Two sides of a triangle are 7 cm and 3 cm. What is the largest integer third side?

  1. 7 − 3 = 4 and 7 + 3 = 10
  2. ∴ 4 cm < third side < 10 cm
  3. Integer options: 5, 6, 7, 8, 9
  4. ∴ the largest integer length is 9 cm

Try it yourself: in triangle XYZ, XY = 8 cm and YZ = 5 cm. The smallest integer for XZ is 4 cm (since 3 < XZ < 13).