Triangle Inequality Theorem

key notes :

Definition:

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Mathematical Representation:

If a triangle has sides a, b, and c, then:

  • a + b > c
  • a + c > b
  • b + c > a

Purpose:

  • Determines whether three given side lengths can form a triangle.
  • Helps in constructing and verifying triangles in geometry.

Examples:

Can a triangle be formed with sides 3 cm, 4 cm, and 8 cm?

  • Check: 3 + 4 = 7 (which is not greater than 8) → Not a triangle.

Can a triangle be formed with sides 5 cm, 7 cm, and 10 cm?

  • Check: 5 + 7 = 12> 10.5 +10 =15>7,7 + 10 = 17>5 → Triangle can be formed.

Real-life Applications:

  • Engineering and construction (ensuring stability in structures).
  • Navigation and map-making.
  • Computer graphics and 3D modeling.

Special Cases:

  • If the sum of two sides is equal to the third side, the points form a straight line, not a triangle.
  • If the sum is less than the third side, a triangle cannot be formed.

If 0≤a≤b≤c, then a, b, and c are the side lengths of a triangle if and only if 

  • a+b>c
  • b+c>a
  • c+a>b

Learn with an example

Can the sides of a triangle have lengths 1, 6, and 9?

First, put the three numbers in order from smallest to largest: a=1, b=6, and c=9.

Now check whether a+b>c. Since 1+6=7, it is false that 1+6>9. So, these are not the side lengths of a triangle.

Can the sides of a triangle have lengths 1, 2, and 3?

First, put the three numbers in order from smallest to largest: a=1, b=2, and c=3.

Now check whether a+b>c. Since 1+2=3, it is false that 1+2>3. So, these are not the side lengths of a triangle.

Can the sides of a triangle have lengths 3, 6, and 8?

First, put the three numbers in order from smallest to largest: a=3, b=6, and c=8.

Now check whether a+b>c. Since 3+6=9, it is true that 3+6>8. So, these are the side lengths of a triangle.

Can the sides of a triangle have lengths 3, 6, and 10?

First, put the three numbers in order from smallest to largest: a=3, b=6, and c=10.

Now check whether a+b>c. Since 3+6=9, it is false that 3+6>10. So, these are not the side lengths of a triangle.

Can the sides of a triangle have lengths 2, 3, and 5?

First, put the three numbers in order from smallest to largest: a=2, b=3, and c=5.

Now check whether a+b>c. Since 2+3=5, it is false that 2+3>5. So, these are not the side lengths of a triangle.

Can the sides of a triangle have lengths 2, 7, and 10?

First, put the three numbers in order from smallest to largest: a=2, b=7, and c=10.

Now check whether a+b>c. Since 2+7=9, it is false that 2+7>10. So, these are not the side lengths of a triangle.

Can the sides of a triangle have lengths 1, 10, and 10?

First, put the three numbers in order from smallest to largest: a=1, b=10, and c=10.

Now check whether a+b>c. Since 1+10=11, it is true that 1+10>10. So, these are the side lengths of a triangle.

Let’s practice!