Triangle Inequality in Real Situations

This matches the Pythagorean triple: 8² + 15² = 64 + 225 = 289 = 17².

So it’s a valid right triangle and also obeys the triangle inequality.

Subtract → 12 − 5 = 7; Add → 12 + 5 = 17.

So the third side must be between 7 and 17.

The two shorter sides equal the third. That’s a straight line, not a triangle.

The student is incorrect.

The third side must be greater than 10 − 10 = 0 and less than 10 + 10 = 20.

15 fits that rule.

All sides are different, and every pair adds to more than the third.

That’s a classic scalene triangle.

Maria wants to build a garden in the shape of a triangle.

To check if it’s possible, we test the triangle rule:

8 + 5 = 13, which is greater than 12, so the sides can form a triangle.

✅ Maria’s plan works!

Here, the rope lengths just equal the longest piece.

Since the sum of the two shorter sides must be greater than, not equal to, the third, these cannot form a triangle.

Let’s find the possible third side.

10 − 7 = 3 and 10 + 7 = 17, so the third side must be between 3 and 17.

A 12 cm side fits perfectly in that range.

John thinks these sides can make a triangle.

But 9 + 12 = 21, which is barely larger than 20.

If the sum were equal or smaller, it wouldn’t close.

This combination doesn’t safely form a triangle.

A builder is making a triangular frame.

The third beam must be greater than |8 − 6| = 2 and less than 8 + 6 = 14.

A 3 m beam works perfectly.

We subtract and add: 14 − 9 = 5, 14 + 9 = 23.

So x must be greater than 5 and less than 23.

When we add the two shorter sides, 10 + 5 = 15, which is less than 16.

That means the sides would not meet – no triangle is possible.

The park designer wants to find a third side.

The difference = 30 − 25 = 5, and the sum = 55.

So the third side must be between 5 and 55.

54 m fits this perfectly.

13 − 5 = 8 and 13 + 5 = 18, so the third side must be between 8 and 18.

The carpenter cannot make a triangle because the two short planks (3 + 4 = 7) don’t reach the length of the longest plank (9).

This is the triangle inequality law that every triangle must follow – always!

Add the two shorter sides: 3 + 6 = 9, which is less than 10.

That fails the triangle rule.

The third side must be greater than 11 − 8 = 3 and less than 11 + 8 = 19.

A 4 cm side fits nicely.

Two sides are equal, and 6 + 6 > 9, so it forms an isosceles triangle.

Here, the sum of the shorter sides is 11, which is just barely enough but not strictly greater than 10.

It doesn’t satisfy the rule, so no triangle can form.