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AMC 10 USA Math Olympiad

Triangle Inequality Theorem – Explanation

The simplest example of power mean inequality is the arithmetic mean – geometric mean inequality. Learn in this self-learning module for math olympiad

What is Triangle Inequality Theorem ?

If I want to give you a perfect definition for Triangle Inequality then I can say : –

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

It follows from the fact that a straight line is the shortest path between two points. The inequality is strict if the triangle is non-degenerate (meaning it has a non-zero area).

Triangle Inequality Theorem

So in other words we can say that : It is not possible to construct a triangle from three line segments if any of them is longer than the sum of the other two. This is known as The Converse of the Triangle Inequality theorem .

So suppose we have three sides lengths as 6 m, 4 m and 3 m then can we draw a triangle with this side ? The answer will be YES we can.

Suppose side a = 3 m

length of side b = 4 m

Length of side c = 6 m

if side a + side b > side c then only we can draw the triangle or

side b + side c > side a or

side a + side c > side b

So from the above example we can find that 4 m + 3 m > 6 m

But look if we try to take 4 m + 6 m \(\geq \) 3 m .

This inequality is particularly useful and shows up frequently on Intermediate level geometry problems. It also provides the basis for the definition of a metric spaces and analysis.

Problem using Triangle Inequality :


In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?

  • 43
  • 44
  • 45
  • 46

Key Concepts


Triangle Inequality

Inequality

Geometry

Check the Answer


The answer is 43 m

AMC – 2006 – 10 B – Problem 10

Secrets in Inequalities.

Try with Hints


This can be a very good example to show Triangle Inequality

Let ‘ x ‘ be the length of the first side of the given triangle. So the length of the second side will be 3 x and that of the third side be 15 . Now apply triangle inequality and try to find the possible values of the sides.

If we apply Triangle Inequality here then the expression will be like

\(3 x < x + 15 \)

\( 2 x < 15 \)

\( x < \frac {15}{2}\)

x < 7.5

Now do the rest of the problem ………..

I am sure you have already got the answer but let me show the rest of the steps for this sum

If x < 7.5 then

The largest integer satisfying this inequality is 7.

So the largest perimeter is 7 + 3.7 +15 = 7 + 21 + 15 = 43.

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