Triangle inequality

Can you make a triangle with sides $2$ cm, $3$ cm, and $10$ cm? Try with sticks , the short sticks cannot stretch far enough. The rule that explains this is the triangle inequality.

Idea

For any three lengths to be the sides of a triangle, each side must be less than the sum of the other two: $a + b > c, \quad b + c > a, \quad c + a > b.$

In practice, you only need to check the two shortest sides against the longest: if $a + b > c$ (with $c$ the longest), the other two inequalities are automatic.

Why? Imagine the longest side fixed on the ground. The two shorter sides hinge at the endpoints. For them to meet above the ground (forming the third vertex), their sum must exceed the length of the side connecting the two hinges. If they are too short, they cannot reach each other.

Equality case. If $a + b = c$ exactly, the triangle "collapses" , the three vertices lie on a straight line. So strict inequality is needed.

The triangle inequality is also a consistency check. When a problem says "the sides are $5, 6, 12$", it has no triangle. Spotting this early saves time.

A useful consequence: in a triangle, the longest side is opposite the largest angle, and vice versa. (Long sides need wide angles to reach far.)

Worked examples

Example 1. Can sides $4, 5, 8$ form a triangle?

Check: $4 + 5 = 9 > 8$ ✓. Other checks: $5+8>4$, $4+8>5$ , both true. So yes.

Example 2. Can sides $3, 4, 7$ form a triangle?

$3 + 4 = 7$. Not greater, equal , so no triangle (it collapses).

Example 3. Two sides of a triangle are $5$ and $9$. What are the possible values of the third side?

Let the third side be $x$. Then $5 + 9 > x$ and $|9 - 5| < x$, i.e., $4 < x < 14$. So $x$ can be any value strictly between $4$ and $14$ , e.g., $5, 6, 7, \dots, 13$ if integers.

Example 4. A triangle has sides $a, a, 2a - 1$. For what values of $a$ does this form a valid triangle?

We need $a + a > 2a - 1 \Rightarrow 2a > 2a - 1$ , always true. And $a + (2a - 1) > a \Rightarrow 2a > 1 \Rightarrow a > 0.5$. So any $a > 0.5$.

Try it yourself

1. Can sides $7, 10, 12$ form a triangle?
2. Can sides $2, 3, 6$ form a triangle?
3. Two sides are $4$ and $11$. Find the range for the third.
4. A triangle has sides $5, x, 8$. What integer values can $x$ take?
5. Show that an equilateral triangle (sides $a, a, a$) always satisfies the inequality.
6. Two sides of an isosceles triangle are $6$ each. The third side?
7. Sides $7, 7, 14$ , triangle or not?
8. Sides $a, b, c$ with $a = b + c$ , what do you get?

Activity

Take a long stick of $24$ cm. Cut it into three pieces. Try various cuts (e.g., $6+8+10$, $5+9+10$, $2+5+17$) and try to form a triangle with each set. Verify the inequality predicts success or failure.