Triangle Inequalities

Triangle Inequalities are fundamental principles governing the relationships between the sides and angles of a triangle. These inequalities ensure that a triangle can actually be formed. Essentially, they state that there are certain conditions that sides and angles must satisfy for a closed three-sided figure to exist.

The core idea revolves around two main concepts:

• Side Length Relationship: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If this condition is not met, the sides will not be able to “close” to form a triangle.
• Angle-Side Relationship: In any triangle, the angle opposite the longer side is larger than the angle opposite the shorter side. Conversely, the side opposite the larger angle is longer than the side opposite the smaller angle.

Formulae

Let $a$, $b$, and $c$ be the lengths of the sides of a triangle. The triangle inequalities are expressed as:

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

Additionally, if $a < b$, then the angle opposite side $a$ (let's call it $A$) is less than the angle opposite side $b$ (let's call it $B$), i.e., $A < B$. And vice versa.

Examples

Here are some examples to illustrate the concepts:

Example-1

Can a triangle be formed with sides of lengths 3, 4, and 5?

Yes, because:

• $3 + 4 > 5$ (7 > 5)
• $3 + 5 > 4$ (8 > 4)
• $4 + 5 > 3$ (9 > 3)

Example-2

Can a triangle be formed with sides of lengths 2, 2, and 5?

No, because:

• $2 + 2 \ngtr 5$ (4 is not greater than 5)

Theorem with Proof

Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Proof:

Consider a triangle $ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively. We will prove $a + b > c$.

Extend side $AC$ (length $b$) to a point $D$ such that $CD = a$. Draw the line segment $BD$.

Since $AD = AC + CD = b + a$.

In $\triangle ABD$, $AB + BD > AD$ (by the triangle inequality, the sum of two sides is greater than the third side).

Also, in $\triangle BCD$, $BD = BC$. But $AD = a+b$, so $AB + BD > a + b$

Hence $a + b > c$.

Common mistakes by students

Students often make the following mistakes:

• Forgetting to check all three inequalities: They might only check one or two inequalities and incorrectly conclude that a triangle can be formed.
• Confusing the inequality signs: Using the wrong inequality sign (e.g., using $\ge$ instead of $>$) when checking the side length conditions.
• Misinterpreting angle-side relationships: Confusing which angle is opposite which side, especially when dealing with diagrams.

Real Life Application

Triangle inequalities are fundamental in various real-life applications:

• Construction: Architects and engineers use triangle inequalities to ensure the structural stability of buildings, bridges, and other structures. The sides of the triangles must satisfy these conditions to prevent collapse.
• Navigation and Surveying: Determining distances and angles in surveying relies on triangle properties, including triangle inequalities.
• Computer Graphics: In rendering 3D objects, triangle meshes are used, and the inequalities are used for checking and validating shapes.

Fun Fact

The triangle inequality also applies in the context of distances in spaces (like the distance between two points). Think about walking from point A to point C. The shortest path is always the straight line. The distance from A to B plus the distance from B to C will always be greater than or equal to the direct distance from A to C.

Recommended YouTube Videos for Deeper Understanding

Q.1 Which of the following sets of side lengths CANNOT form a triangle?

Check Solution

Ans: C

Check if the sum of any two sides is greater than the third side. For C, 2 + 2 is not greater than 5.

Q.2 In triangle $ABC$, $AB = 7$, $BC = 10$. Which of the following could be the length of $AC$?

Check Solution

Ans: D

According to the triangle inequality, $10-7 < AC < 10+7$, thus $3 < AC < 17$. Only 15 satisfies.

Q.3 In triangle $PQR$, if $\angle P = 60^\circ$ and $\angle Q = 70^\circ$, which side is the longest?

Check Solution

Ans: B

First find the third angle: $\angle R = 180^\circ – 60^\circ – 70^\circ = 50^\circ$. The largest angle is $\angle Q = 70^\circ$, so the longest side is opposite it, which is $PR$.

Q.4 The sides of a triangle are in the ratio 3:4:5. If the perimeter of the triangle is 36, what is the length of the longest side?

Check Solution

Ans: C

Let the sides be $3x$, $4x$, and $5x$. The perimeter is $3x + 4x + 5x = 12x$. $12x = 36$, so $x = 3$. The longest side is $5x = 5(3) = 15$.

Q.5 If two sides of a triangle are 8 cm and 15 cm, the length of the third side must be between:

Check Solution

Ans: B

Using the triangle inequality, the third side must be greater than $15-8=7$ and less than $15+8=23$.

Next Topic: Angle Sum Property of a Quadrilateral