Inequalities in Triangles

Q: Can a triangle have sides 1 cm, 2 cm, and 3 cm?

No. 1 + 2 = 3, which is NOT greater than 3. The triangle inequality requires strict inequality (>), not equality (≥). These lengths form a straight line, not a triangle.

Class 9Triangles

Inequalities in triangles describe the relationship between the sides and angles of a triangle. These theorems establish that in any triangle, the longer side is opposite the larger angle, and the sum of any two sides is always greater than the third.

These results are fundamental to understanding triangle geometry. They help determine whether a set of three lengths can form a triangle, and which side is opposite which angle.

This topic is part of NCERT Class 9 Mathematics, Chapter: Triangles. The key theorems covered are the angle-side relationship, its converse, and the triangle inequality theorem.

What is Inequalities in Triangles?

Key Theorems:

Theorem 1 (Angle-Side Relationship):

If two sides of a triangle are unequal, the angle opposite the longer side is larger.

Theorem 2 (Converse):

If two angles of a triangle are unequal, the side opposite the larger angle is longer.

Theorem 3 (Triangle Inequality):

The sum of any two sides of a triangle is greater than the third side.

Important:

In a triangle with sides a, b, c: a + b > c, b + c > a, and a + c > b.
The largest angle is opposite the longest side.
The smallest angle is opposite the shortest side.
If any one of the three triangle inequality conditions fails, no triangle can be formed.

Inequalities in Triangles Formula

Key Results:

1. Angle-Side Relationship:

If AB > AC in triangle ABC, then ∠C > ∠B.
The larger side is opposite the larger angle.

2. Side-Angle Relationship (Converse):

If ∠C > ∠B in triangle ABC, then AB > AC.
The larger angle has the longer side opposite to it.

3. Triangle Inequality Theorem:

a + b > c, b + c > a, a + c > b

All three conditions must hold simultaneously.
Sufficient to check that the sum of the two smaller sides > the largest side.

4. Corollary:

The difference of any two sides < the third side.
|a − b| < c < a + b

5. Shortest distance:

Of all line segments from a point to a line, the perpendicular is the shortest.

Derivation and Proof

Proof of Theorem 1: If two sides of a triangle are unequal, the angle opposite the longer side is larger.

Given: Triangle ABC with AB > AC.

To prove: ∠ACB > ∠ABC.

Construction: Mark point D on AB such that AD = AC.

Proof:

In triangle ACD: AD = AC, so ∠ACD = ∠ADC (base angles of isosceles triangle).
Since D lies on AB (between A and B), ∠ADC is an exterior angle of triangle BCD.
Exterior angle > each interior opposite angle: ∠ADC > ∠DBC (i.e., ∠ADC > ∠ABC).
Since ∠ACD = ∠ADC, we get ∠ACD > ∠ABC.
Since ∠ACB > ∠ACD (D is inside the triangle), we get ∠ACB > ∠ACD > ∠ABC.
Therefore: ∠ACB > ∠ABC. Proved.

Proof of Triangle Inequality Theorem:

Given: Triangle ABC.

To prove: AB + AC > BC.

Construction: Extend BA to point D such that AD = AC. Join DC.

Proof:

In triangle ACD: AD = AC, so ∠ACD = ∠ADC.
∠BCD > ∠ACD (since ∠BCD = ∠BCA + ∠ACD).
Therefore ∠BCD > ∠ADC, which means ∠BCD > ∠BDC.
In triangle BCD, the side opposite the larger angle is longer: BD > BC.
BD = BA + AD = BA + AC.
Therefore: BA + AC > BC. Proved.

Types and Properties

Types of triangle inequality problems:

1. Checking if a triangle can be formed

Given three lengths, verify that the sum of any two > the third.
Sufficient to check: (smallest + middle) > largest.

2. Finding the range of the third side

Given two sides a and b (a > b), the third side c satisfies: (a − b) < c < (a + b).

3. Ordering sides given angles

Given the angles, determine which side is longest, shortest.
Larger angle → longer opposite side.

4. Ordering angles given sides

Given the side lengths, determine which angle is largest.
Longer side → larger opposite angle.

5. Shortest path problems

The perpendicular from a point to a line is the shortest distance.
Applications in routing and network design.

6. Proof-based problems

Prove specific inequalities using the three main theorems.

Solved Examples

Example 1: Example 1: Checking triangle formation

Problem: Can a triangle be formed with sides 5 cm, 8 cm, and 15 cm?

Solution:

Check triangle inequality:

5 + 8 = 13 < 15 ✗

The sum of the two smaller sides (13) is NOT greater than the largest side (15).

Answer: No, a triangle cannot be formed.

Example 2: Example 2: Valid triangle check

Problem: Can a triangle be formed with sides 7 cm, 10 cm, and 12 cm?

Solution:

Check:

7 + 10 = 17 > 12 ✓
7 + 12 = 19 > 10 ✓
10 + 12 = 22 > 7 ✓

All three conditions are satisfied.

Answer: Yes, a triangle can be formed.

Example 3: Example 3: Range of the third side

Problem: Two sides of a triangle are 8 cm and 13 cm. Find the range of possible values for the third side.

Solution:

Using the inequality:

Third side > |13 − 8| = 5 cm
Third side < 13 + 8 = 21 cm

Answer: The third side must be between 5 cm and 21 cm (exclusive): 5 < c < 21.

Example 4: Example 4: Ordering sides from angles

Problem: In triangle ABC, ∠A = 70°, ∠B = 50°, ∠C = 60°. Arrange the sides in ascending order.

Solution:

Ordering angles: ∠B < ∠C < ∠A (50° < 60° < 70°)

By Theorem 2: Side opposite larger angle is longer.

Side opposite ∠B = AC (shortest)
Side opposite ∠C = AB (middle)
Side opposite ∠A = BC (longest)

Answer: AC < AB < BC

Example 5: Example 5: Ordering angles from sides

Problem: In triangle PQR, PQ = 9 cm, QR = 5 cm, PR = 7 cm. Arrange the angles in descending order.

Solution:

Ordering sides: QR < PR < PQ (5 < 7 < 9)

By Theorem 1: Angle opposite longer side is larger.

∠R (opposite PQ = 9) is the largest
∠Q (opposite PR = 7) is the middle
∠P (opposite QR = 5) is the smallest

Answer: ∠R > ∠Q > ∠P

Example 6: Example 6: Right triangle inequality

Problem: In a right triangle, the hypotenuse is 10 cm and one leg is 6 cm. Verify the triangle inequality.

Solution:

Finding the other leg:

b = √(10² − 6²) = √(100 − 36) = √64 = 8 cm

Checking:

6 + 8 = 14 > 10 ✓
6 + 10 = 16 > 8 ✓
8 + 10 = 18 > 6 ✓

Answer: Triangle inequality is satisfied. The hypotenuse is the longest side (opposite the 90° angle).

Example 7: Example 7: Multiple choice - which set cannot form a triangle?

Problem: Which of the following sets of lengths CANNOT form a triangle? (a) 3, 4, 5 (b) 2, 3, 6 (c) 5, 6, 7 (d) 8, 10, 12

Solution:

(a) 3 + 4 = 7 > 5 ✓
(b) 2 + 3 = 5 < 6 ✗
(c) 5 + 6 = 11 > 7 ✓
(d) 8 + 10 = 18 > 12 ✓

Answer: (b) 2, 3, 6 cannot form a triangle.

Example 8: Example 8: Proving an inequality

Problem: Prove that in any triangle ABC, AB + BC > AC.

Solution:

This is the triangle inequality theorem.

Proof:

Extend BA to D such that AD = AC. Join DC.
In triangle ACD: AD = AC, so ∠ACD = ∠ADC.
∠BCD = ∠BCA + ∠ACD > ∠ACD = ∠ADC = ∠BDC.
In triangle BCD: ∠BCD > ∠BDC, so BD > BC (side opposite larger angle is longer).
BD = BA + AD = BA + AC (since AD = AC).
Therefore: BA + AC > BC, i.e., AB + AC > BC. Proved.

Example 9: Example 9: Shortest distance to a line

Problem: From point P, lines PA, PB, and PC are drawn to line l, where PB ⊥ l. PA = 7 cm and PB = 5 cm. Which is shorter, PA or PB? Why?

Solution:

PB is perpendicular to line l.
By the theorem: the perpendicular from a point to a line is the shortest segment.
PA (oblique) = 7 cm > PB (perpendicular) = 5 cm.

Answer: PB is shorter because the perpendicular distance is always the shortest distance from a point to a line.

Example 10: Example 10: Integer side lengths

Problem: Two sides of a triangle are 5 cm and 9 cm. How many triangles with integer third side are possible?

Solution:

Range of third side c:

|9 − 5| < c < 9 + 5
4 < c < 14

Integer values: c = 5, 6, 7, 8, 9, 10, 11, 12, 13

Answer: 9 triangles are possible (with integer sides).

Real-World Applications

Applications of Triangle Inequalities:

Construction: Verifying whether three beams of given lengths can form a triangular framework.
Navigation: The triangle inequality proves that a straight line (direct path) is always shorter than any detour through a third point.
Network routing: Shortest path algorithms use triangle inequality to optimise routes.
Surveying: Checking consistency of measurements — if three measured distances violate the triangle inequality, there is a measurement error.
Computer graphics: Validating triangle meshes in 3D models.
Optimisation: Many mathematical proofs and optimisation algorithms rely on the triangle inequality as a fundamental constraint.

Key Points to Remember

In a triangle, the side opposite the larger angle is always longer.
In a triangle, the angle opposite the longer side is always larger.
Triangle inequality: The sum of any two sides must be greater than the third side.
If a + b ≤ c for any arrangement, no triangle can be formed.
The third side c must satisfy: |a − b| < c < a + b.
The largest angle is opposite the longest side; the smallest angle is opposite the shortest side.
In a right triangle, the hypotenuse is the longest side (opposite 90°).
The perpendicular from a point to a line is the shortest distance.
The difference of two sides is always less than the third side.
To check if three lengths form a triangle, it suffices to verify (smallest + middle) > largest.

Practice Problems

Can a triangle be formed with sides 4 cm, 6 cm, and 11 cm? Justify.
Two sides of a triangle are 7 cm and 15 cm. Find the range of the third side.
In triangle ABC, ∠A = 45°, ∠B = 75°, ∠C = 60°. Arrange the sides in descending order.
In triangle DEF, DE = 11 cm, EF = 6 cm, DF = 8 cm. Which is the largest angle?
Two sides of a triangle are 10 cm and 14 cm. How many triangles with integer third side are possible?
Prove that in any triangle, the sum of any two sides is greater than the third side.
In a right triangle with legs 5 cm and 12 cm, verify the triangle inequality.
ABC is a triangle with AB = 6 cm, BC = 9 cm, AC = 4 cm. List the angles in increasing order.

Frequently Asked Questions

Q1. What is the triangle inequality theorem?

The triangle inequality 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 a, b, c are the sides: a + b > c, b + c > a, and a + c > b.

Q2. How do you check if three lengths can form a triangle?

Check that the sum of the two smaller lengths is greater than the largest. If the smallest + middle > largest, a triangle can be formed. If not, it cannot.

Q3. Which side is opposite the largest angle?

The longest side is always opposite the largest angle. For example, in a right triangle, the hypotenuse (longest side) is opposite the right angle (largest angle).

Q4. What is the range of the third side of a triangle?

If two sides are a and b (with a > b), the third side c must satisfy: (a − b) < c < (a + b). The third side is strictly between the difference and the sum.

Q5. Can a triangle have sides 1 cm, 2 cm, and 3 cm?

No. 1 + 2 = 3, which is NOT greater than 3. The triangle inequality requires strict inequality (>), not equality (≥). These lengths form a straight line, not a triangle.

Q6. What is the shortest distance from a point to a line?

The perpendicular from the point to the line is the shortest distance. Any other line segment from the point to the line (oblique) will be longer.

Q7. Is triangle inequality in the CBSE Class 9 syllabus?

Yes. Inequalities in a triangle, including the angle-side relationship, its converse, and the triangle inequality theorem are part of CBSE Class 9 Mathematics, Chapter: Triangles.

Q8. How are angle-side relationships used in problems?

Given the angles, you can order the sides (larger angle means longer opposite side). Given the sides, you can order the angles (longer side means larger opposite angle). This helps in comparing and ranking without measuring.

Q9. Can a triangle have two obtuse angles?

No. If two angles were each greater than 90 degrees, their sum alone would exceed 180 degrees, which violates the angle sum property. A triangle can have at most one obtuse angle.

Q10. What is the Exterior Angle Inequality?

An exterior angle of a triangle is always greater than either of the two non-adjacent (remote) interior angles. This follows from the fact that the exterior angle equals the sum of the two remote interior angles.

Inequalities in Triangles

What is Inequalities in Triangles?

Inequalities in Triangles Formula

Derivation and Proof

Types and Properties

Solved Examples

Example 1: Example 1: Checking triangle formation

Example 2: Example 2: Valid triangle check

Example 3: Example 3: Range of the third side

Example 4: Example 4: Ordering sides from angles

Example 5: Example 5: Ordering angles from sides

Example 6: Example 6: Right triangle inequality

Example 7: Example 7: Multiple choice - which set cannot form a triangle?

Example 8: Example 8: Proving an inequality

Example 9: Example 9: Shortest distance to a line

Example 10: Example 10: Integer side lengths

Real-World Applications

Key Points to Remember

Practice Problems

Frequently Asked Questions

Q1. What is the triangle inequality theorem?

Q2. How do you check if three lengths can form a triangle?

Q3. Which side is opposite the largest angle?

Q4. What is the range of the third side of a triangle?

Q5. Can a triangle have sides 1 cm, 2 cm, and 3 cm?

Q6. What is the shortest distance from a point to a line?

Q7. Is triangle inequality in the CBSE Class 9 syllabus?

Q8. How are angle-side relationships used in problems?

Q9. Can a triangle have two obtuse angles?

Q10. What is the Exterior Angle Inequality?

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