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


But try the problem first…

The answer is 43 m

Source
Suggested Reading

AMC – 2006 – 10 B – Problem 10

Secrets in Inequalities.

Try with Hints


First Hint……..

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 you really need another hint try this out …………….

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 ………..

Final Step

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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