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.
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:
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
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.