Triangle Inequality

Triangles are mostly used in civil constructions, surveying and commonly in pyramids. Triangle is the most stable shape out of all the figures. Do you know why? Triangle inequality is a mathematical principle that deals with the sides of a triangle. It helps to calculate the unknown side of a triangle.

Definition

According to the triangle inequality theorem, the sum of two sides of a triangle must be greater than the side. For example, consider a triangle ABC with sides AB, BC and AC.
Then according to the triangle inequality theorem:

• AB + BC must be greater than AC, or AB + BC > AC.
• AB + AC must be greater than BC, or AB + AC > BC
• BC + AC must be greater than AB, or BC + AC > AB.

Working of the triangle inequality

We can find the application of triangle inequality in practical life situations. For example, consider you wish to cross a triangle path without going from its centre. You have to cross the path from the sides only. Then which side will you choose? Of course, the sides have a shorter route. This way, the triangular theorem comes in handy in practical life. Let us study how to construct a triangle by knowing the sides of a triangle.

1. Construct a line segment of length AB with some units, say 4 cm.
2. Next, using a compass, construct an arc of any length from point A towards the top. The length of this side can be anything.
3. Now, repeat the step and construct an arc from point B of any length. This arc will cut the previous arc at a particular point. Name this point as C.
4. Join AC and BC with a ruler.
5. Measure the sides of AC and BC and note them down. We already know the side of AB = 4 cm.
6. After measuring the sides, find the sum of sides AB + BC, AB + AC and BC + AC.
7. After doing so, these conditions must be satisfied to prove the triangular inequality theorem:

AB + BC must be greater than AC, or AB + BC > AC
AB + AC must be greater than BC, or AB + AC > BC
BC + AC must be greater than AB, or BC + AC > AB

Proof of triangle inequality

Consider a triangle ABC. According to the triangle inequality theorem, we need to prove AB + AC > BC.

To prove the theorem, we need to extend BA to a point D such that AC = AD. Join C to D.

We must note that angle ACD is equal to angle D. This implies that in triangle BCD, angle BCD > angle D (sides opposite larger angles are larger). Thus, BD > BC.

AB + AD > BC

AB + AC > BC (because AD = AC)

Hence, proved that the sum of the other two sides of a triangle is greater than the third side.

We can also draw additional conclusions from the above theorem-proof:

1. If the sum of any two sides of a triangle is greater than the third side, then the difference of any two sides of a triangle will be less than the third side.
2. The side opposite to the largest angle is the largest side as well.