Triangle Inequality Property
Can any three lengths form a triangle? The answer is no. There is a rule that decides whether three given lengths can form a triangle. This rule is called the Triangle Inequality Property.
The Triangle Inequality Property states that the sum of any two sides of a triangle is always greater than the third side. If this condition is not satisfied, the three lengths cannot form a triangle.
This property is covered in the NCERT Class 7 Triangles chapter. It is useful for checking whether a triangle is possible before attempting to draw or construct one.
What is Triangle Inequality Property?
Definition: The Triangle Inequality Property states that in any triangle, the sum of the lengths of any two sides is always greater than the length of the third side.
If the three sides of a triangle are a, b, and c, then ALL three conditions must hold:
- a + b > c
- b + c > a
- a + c > b
Key Points:
- ALL three conditions must be true for a triangle to be possible.
- If even ONE condition fails, the triangle cannot exist.
- If a + b = c (equal, not greater), the three points are collinear (they lie on a straight line) and do NOT form a triangle.
- This property applies to ALL types of triangles: scalene, isosceles, equilateral, right-angled, etc.
Triangle Inequality Property Formula
Triangle Inequality Property:
Sum of any two sides > Third side
For sides a, b, c:
- a + b > c
- b + c > a
- a + c > b
Quick Check Method:
In practice, you only need to check whether the sum of the two shorter sides is greater than the longest side. If this condition holds, the other two conditions automatically hold.
Shortest side + Middle side > Longest side
Related Property: The difference of any two sides is always less than the third side.
- |a - b| < c
- |b - c| < a
- |a - c| < b
Combining both: |a - b| < c < a + b
Derivation and Proof
Why does this property hold?
Think of it physically. Take three sticks of lengths a, b, and c. Try to form a triangle.
- If the two shorter sticks together are shorter than the longest stick, they cannot reach each other to close the triangle. No triangle is possible.
- If the two shorter sticks exactly equal the longest stick, they lie flat along the longest stick forming a straight line. No triangle is formed.
- Only when the two shorter sticks are longer than the longest stick can they bend inward and meet, closing the triangle.
Proof using the shortest path property:
- In geometry, the shortest distance between two points is a straight line.
- In triangle ABC, the straight-line distance from A to C is the side AC.
- Going from A to B and then B to C is a longer path (not straight).
- Therefore: AB + BC > AC.
- This argument works for any pair of sides.
Verification with specific values:
- Sides 3, 4, 5: 3 + 4 = 7 > 5, 4 + 5 = 9 > 3, 3 + 5 = 8 > 4. Triangle possible.
- Sides 1, 2, 5: 1 + 2 = 3 < 5. Triangle NOT possible.
- Sides 3, 3, 6: 3 + 3 = 6 = 6 (not greater). Triangle NOT possible (points are collinear).
Types and Properties
Problems on the triangle inequality property:
1. Checking if a triangle is possible:
- Given three sides, check if the sum of the two smaller sides > the largest side.
2. Finding the range of the third side:
- Given two sides, find the possible range of values for the third side.
- If two sides are a and b, then: |a - b| < third side < a + b.
3. Finding the longest/shortest possible side:
- Given constraints, determine the maximum or minimum possible length of a side.
4. Integer side problems:
- How many triangles with integer sides are possible given certain conditions?
5. Application to real-world problems:
- Checking if given measurements can form a triangular frame, plot, etc.
Solved Examples
Example 1: Example 1: Can these sides form a triangle?
Problem: Can sides of lengths 4 cm, 6 cm, and 9 cm form a triangle?
Solution:
Check: Sum of two smaller sides > largest side?
- 4 + 6 = 10 > 9? Yes.
Let us verify all three conditions:
- 4 + 6 = 10 > 9 ✔
- 6 + 9 = 15 > 4 ✔
- 4 + 9 = 13 > 6 ✔
Answer: Yes, these sides can form a triangle.
Example 2: Example 2: Triangle not possible
Problem: Can sides of lengths 2 cm, 3 cm, and 7 cm form a triangle?
Solution:
Check: Sum of two smaller sides > largest side?
- 2 + 3 = 5 > 7? No, 5 < 7.
The condition fails.
Answer: No, these sides cannot form a triangle. The two shorter sides (2 and 3) cannot reach each other across the longest side (7).
Example 3: Example 3: Equal sum (degenerate case)
Problem: Can sides 5 cm, 5 cm, and 10 cm form a triangle?
Solution:
- 5 + 5 = 10. This is equal to the third side, not greater.
- The condition requires strictly greater than.
- When sum = third side, the three points lie on a straight line (degenerate triangle).
Answer: No, these sides cannot form a triangle.
Example 4: Example 4: Finding the range of the third side
Problem: Two sides of a triangle are 5 cm and 8 cm. Find the range of possible values for the third side.
Solution:
Let the third side = x.
Using the triangle inequality:
- 5 + 8 > x, so x < 13
- 5 + x > 8, so x > 3
- 8 + x > 5, so x > -3 (always true for positive lengths)
Combining:
- 3 < x < 13
The third side must be greater than 3 cm and less than 13 cm.
Answer: The third side can be any value between 3 cm and 13 cm (exclusive).
Example 5: Example 5: Integer values for the third side
Problem: Two sides of a triangle are 6 cm and 10 cm. How many integer values are possible for the third side?
Solution:
Range:
- |10 - 6| < third side < 10 + 6
- 4 < third side < 16
Integer values: 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15
Count: 11 values
Answer: There are 11 possible integer values for the third side.
Example 6: Example 6: Equilateral triangle check
Problem: Verify the triangle inequality for an equilateral triangle with side 7 cm.
Solution:
- 7 + 7 = 14 > 7 ✔
- 7 + 7 = 14 > 7 ✔
- 7 + 7 = 14 > 7 ✔
All three conditions are satisfied.
Answer: An equilateral triangle with side 7 cm satisfies the triangle inequality property.
Example 7: Example 7: Checking multiple sets of sides
Problem: Which of these can form a triangle? (a) 3, 4, 5 (b) 1, 1, 3 (c) 5, 5, 5 (d) 2, 6, 3
Solution:
- (a) 3 + 4 = 7 > 5. Yes, triangle possible. (It is a right triangle.)
- (b) 1 + 1 = 2 < 3. No, triangle not possible.
- (c) 5 + 5 = 10 > 5. Yes, triangle possible. (Equilateral triangle.)
- (d) 2 + 3 = 5 < 6. No, triangle not possible.
Answer: Only (a) and (c) can form triangles.
Example 8: Example 8: Longest side of a triangle
Problem: The two shorter sides of a triangle are 7 cm and 11 cm. What is the maximum possible integer length of the third side?
Solution:
- Third side < 7 + 11 = 18
- Maximum integer value = 17 cm
Verification: 7 + 11 = 18 > 17 ✔. And 7 + 17 = 24 > 11 ✔. And 11 + 17 = 28 > 7 ✔.
Answer: The maximum integer length is 17 cm.
Example 9: Example 9: Relationship between sides and angles
Problem: In a triangle with sides 5 cm, 12 cm, and 13 cm, which side is opposite the largest angle?
Solution:
- The longest side = 13 cm.
- In any triangle, the largest angle is opposite the longest side.
- Therefore, the largest angle is opposite the side of 13 cm.
Note: 5² + 12² = 25 + 144 = 169 = 13², so this is a right triangle. The right angle (90°) is opposite the side of 13 cm (the hypotenuse).
Answer: The side of 13 cm is opposite the largest angle.
Example 10: Example 10: Real-life application
Problem: Rahul wants to build a triangular garden with fences of lengths 4 m, 5 m, and 10 m. Is this possible?
Solution:
- Check: sum of two shorter sides > longest side?
- 4 + 5 = 9 > 10? No, 9 < 10.
The two shorter fences (4 m and 5 m) cannot reach each other across the longest fence (10 m).
Answer: No, Rahul cannot build a triangular garden with these fence lengths. He needs to choose different lengths.
Real-World Applications
Construction: Builders check whether beams of given lengths can form triangular supports. If the triangle inequality is not satisfied, the structure cannot be built.
Navigation: The triangle inequality relates to the shortest path principle. The direct distance between two points is always less than going via a third point.
Network Design: In computer networks and road networks, the triangle inequality ensures that the direct route is shorter than the indirect route through a third node.
Practical Geometry: Before constructing a triangle with given measurements (SSS construction), you must first verify that the triangle inequality is satisfied. Otherwise, the construction will fail.
GPS and Mapping: Triangle inequality is used in distance calculations and error checking in GPS positioning systems.
Key Points to Remember
- The sum of any two sides of a triangle is always greater than the third side.
- All three conditions must hold: a + b > c, b + c > a, and a + c > b.
- Quick check: Sum of the two shorter sides > longest side.
- If the sum equals the third side, the points are collinear (no triangle).
- If the sum is less than the third side, no triangle is possible.
- The difference of any two sides is always less than the third side.
- Range of third side: |a - b| < c < a + b.
- The longest side of a triangle is opposite the largest angle.
- The shortest side of a triangle is opposite the smallest angle.
- This property applies to ALL triangles: scalene, isosceles, equilateral, right-angled.
Practice Problems
- Can sides 7 cm, 10 cm, and 5 cm form a triangle? Justify.
- Can sides 1 cm, 2 cm, and 3 cm form a triangle?
- Two sides of a triangle are 9 cm and 14 cm. Find the range of the third side.
- How many triangles with integer sides can be formed if two sides are 4 cm and 7 cm?
- The three sides of a triangle are x, x + 1, and x + 2 cm. Find the minimum integer value of x.
- Can an isosceles triangle have sides 3 cm, 3 cm, and 8 cm?
Frequently Asked Questions
Q1. What is the triangle inequality property?
The triangle inequality states that the sum of any two sides of a triangle must be strictly greater than the third side. All three such conditions must hold for a triangle to exist.
Q2. Why must the sum be 'greater than' and not 'equal to'?
If the sum equals the third side, the three points lie on a straight line (degenerate case). There is no enclosed area, so no triangle is formed.
Q3. Do I need to check all three conditions?
In practice, you only need to check one condition: sum of the two SHORTER sides > longest side. If this holds, the other two conditions automatically hold.
Q4. What is the range of the third side?
If two sides are a and b, the third side c must satisfy: |a - b| < c < a + b. For example, if two sides are 5 and 8, then 3 < c < 13.
Q5. Does this property apply to right triangles?
Yes. It applies to ALL triangles. For a right triangle with sides 3, 4, 5: 3 + 4 = 7 > 5 (satisfied). The hypotenuse is always less than the sum of the other two sides.
Q6. Can a triangle have sides 0 cm?
No. All sides of a triangle must be positive (greater than 0). A side of length 0 would mean two vertices coincide, which does not form a triangle.
Q7. What happens if two sides are equal?
The triangle inequality still applies. For an isosceles triangle with sides a, a, b: the condition is a + a > b, i.e., 2a > b, i.e., b < 2a. The third side must be less than twice the equal side.
Q8. How is this used in triangle construction?
Before constructing a triangle using SSS (given three sides), verify the triangle inequality. If 3 + 4 = 7 > 5, construction is possible. If 2 + 3 = 5 < 8, construction will fail.
Q9. Which side is opposite the largest angle?
The longest side is always opposite the largest angle. This is a related property: in any triangle, larger sides are opposite larger angles.
Q10. Can a triangle have two sides that together are much greater than the third?
Yes. For example, sides 10, 10, 1 satisfy the triangle inequality (10 + 10 = 20 > 1). This forms a very flat (nearly degenerate) isosceles triangle.
Related Topics
- Angle Sum Property of Triangle
- Classification of Triangles
- Exterior Angle Property of Triangle
- Pythagoras Theorem
- Properties of Isosceles Triangle
- Properties of Equilateral Triangle
- Medians and Altitudes of Triangle
- Right-Angled Triangle Property
- Congruent Triangles - Proofs
- Inequalities in Triangles
- Similar Triangles
- Basic Proportionality Theorem (BPT)
- Converse of Basic Proportionality Theorem
- Criteria for Similarity of Triangles
