Triangle Inequality Theorem

key notes :

🟢 Triangle Inequality Theorem

🔺 Definition:

The sum of the lengths of any two sides of a triangle is always greater than the length of the third side. Formula:
If a triangle has sides a, b, c, then:

a + b > c

b + c > a

a + c > b

🔹 Purpose:

Helps to check whether three given lengths can form a triangle.

🔹 Example:

If sides are 3 cm, 4 cm, 8 cm → Can they form a triangle?

3 + 4 = 7 ❌ (not > 8)

✅ So, no triangle is possible.

🔹 Important Points:

⚠️ The sum of two sides must be strictly greater, not equal.

✅ Always check all three combinations.

🔄 Works for all types of triangles: scalene, isosceles, equilateral.

🌟 Mnemonic Tip:

“Any two sides together must be bigger than the third side” 👫➕👫 > 👬

🖌️ Visual Representation:

Draw a triangle and label sides a, b, c.

Show arrows with a + b > c, etc. to remember the theorem easily.

💡 Real-life Example:

Building a triangular roof 🏠

Connecting supports in a triangular frame 🏗️

Ensures structure is stable and possible.

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!