In this article, we will discover what the triangle inequality theorem is, just how to use the theorem, and lastly, what reverse triangle inequality entails. In ~ this point, most of us are acquainted with the reality that a triangle has actually three sides.

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The three sides of a triangle are formed when three different line segments sign up with at the vertices that a triangle. In a triangle, we use the tiny letters a, b and also c to signify a triangle’s sides.

In most cases, letter a and b are provided to represent the very first two short sides that a triangle, whereas letter c is provided to stand for the longest side.

What is Triangle Inequality Theorem?

As the name suggests, the triangle inequality theorem is a statement that explains the relationship between the three sides that a triangle. According to the triangle inequality theorem, the sum of any kind of two sides of a triangle is better than or equal to the 3rd side that a triangle.

This statement have the right to symbolically be represented as;

a + b > ca + c > bb + c > a

Therefore, a triangle inequality theorem is a useful tool for checking whether a given collection of three dimensions will form a triangle or not. Merely put, it will certainly not form a triangle if the over 3 triangle inequality conditions are false.

Let’s take it a look in ~ the adhering to examples:

Example 1

Check whether it is feasible to kind a triangle v the adhering to measures:

4 mm, 7 mm, and 5 mm.

Solution

Let a = 4 mm. B = 7 mm and c = 5 mm. Now use the triangle inequality theorem.

a + b > c

⇒ 4 + 7 > 5

⇒ 11> 5 ……. (true)

a + c > b

⇒ 4 + 5 > 7

⇒ 9 > 7…………. (true)

b + c > a

⇒7 + 5 > 4

⇒12 > 4 ……. (true)

Since every three problems are true, that is possible to kind a triangle v the given measurements.

Example 2

Given the measurements; 6 cm, 10 cm, 17 cm. Inspect if the three dimensions can type a triangle.

Solution

Let a = 6 cm, b = 10 cm and c = 17 cm

By triangle inequality theorem, we have;

a + b > c

⇒ 6 + 10 > 17

⇒ 16 > 17 ………. (false, 17 is not much less than 16)

a + c > b

⇒ 6 + 17 > 10

⇒ 23 > 10…………. (true)

b + c > a

10 + 17 > 6

17 > 6 ………. (true)

Since one of the conditions is false, therefore, the three dimensions cannot form a triangle.

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Example 3

Find the feasible values that x because that the triangle displayed below.

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The turning back triangle inequality organize is offered by;

|PQ|>||PR|-|RQ||, |PR|>||PQ|-|RQ|| and |QR|>||PQ|-|PR||

Proof:

|PQ| + |PR| > |RQ| // Triangle Inequality Theorem|PQ| + |PR| -|PR| > |RQ|-|PR| // (i) subtracting the same amount from both next maintains the inequality|PQ| > |RQ| – |PR| = ||PR|-|RQ|| // (ii), properties of pure value|PQ| + |PR| – |PQ| > |RQ|-|PQ| // (ii) individually the same amount from both next maintains the inequality|PR| > |RQ|-|PQ| = ||PQ|-|RQ|| // (iv), nature of pure value|PR|+|QR| > |PQ| //Triangle Inequality Theorem|PR| + |QR| -|PR| > |PQ|-|PR| // (vi) subtracting the same amount from both next maintains the inequality|QR| > |PQ| – |PR| = ||PQ|-|PR|| // (vii), properties of absolute value