Explainer: Inequality in One Triangle: Sides

In this explainer, we will learn how to apply the triangle inequality to determine whether three given lengths can be sides of a triangle.

The Triangle Inequality

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

This leads to three inequalities.

When constructing a triangle with a ruler and a pair of compasses, you need to draw the base of the triangle first. Then, you draw a circle centered at one of the base’s endpoints of a radius the length of the second side, and finally a second circle centered at the other base’s endpoint of a radius the length of the third side. The third vertex of the triangle is then given by the intersection of both circles, as shown in the diagram.

How to Understand the Triangle Inequality

Let’s try to construct triangles of sides 7 cm, 4 cm, and 2 cm. By drawing the 7 cm long side first and then the two circles of radii 4 cm and 2 cm, we get the following diagram.

The two circles do not intersect because 4+2<7. This means that a triangle of sides 7 cm, 4 cm, and 2 cm does not exist.

Note that it does not matter which side you draw as a base, as shown here. Of course, you still cannot construct any triangle.

Example 1: Using the Triangle Inequality

Is it possible to form a triangle with side lengths 3 inches, 5 inches, and 7 inches?

Answer

We need to check that each of the triangle inequalities is true with these side lengths.

  • Is 3+5>7? Yes.
  • Is 5+7>3? Yes.
  • Is 3+7>5? Yes.

Therefore, it is possible to form a triangle with side lengths 3 inches, 5 inches, and 7 inches.

Example 2: Using the Triangle Inequality

Is it possible to form a triangle with side lengths 6 m, 7 m, and 18 m?

Answer

We need to check that each of the triangle inequalities is true with these side lengths.

  • Is 6+7>18? No.

Therefore, it is not possible to form a triangle with side lengths 6 m, 7 m, and 18 m.

Note that as soon as we find that one of the triangle inequalities is not true, we know that it is not possible to form the triangle. We do not need to check the other triangle inequalities.

Example 3: Using the Triangle Inequality

Two sides of a triangle are 5 cm and 8 cm. What is the range of values for the third side?

Answer

We are given two side lengths of a triangle, and we are asked about the range of values for the third side’s length. Let the length of the third side be π‘₯ cm. Let’s now use the triangle inequalities to find the range of values for π‘₯.

  1. π‘₯+5>8. Taking away 5 from each side, we get π‘₯>3.
  2. π‘₯+8>5. Taking away 8 from each side, we get π‘₯>βˆ’3. This does not give us any new information as π‘₯ is a length in centimeters and is thus positive. Furthermore, we know from (1) that π‘₯>3.
  3. 5+8>π‘₯; that is 13>π‘₯, or π‘₯<13.

By combining (1) and (3), we get 3<π‘₯<13.

Example 4: Using the Triangle Inequality

Given that π‘₯ is an integer, determine the possible values of π‘₯.

Answer

We are going to use the three triangle inequalities to find the range of values for π‘₯.

  1. 3π‘₯βˆ’1+π‘₯+3>5π‘₯βˆ’3; by adding the like terms, we get 4π‘₯+2>5π‘₯βˆ’3; by adding 3 to each side, we get 4π‘₯+5>5π‘₯; and by taking away 4π‘₯ from each side, we get 5>π‘₯, or π‘₯<5.
  2. π‘₯+3+5π‘₯βˆ’3>3π‘₯βˆ’1; by adding the like terms, we get 6π‘₯>3π‘₯βˆ’1; by subtracting 3π‘₯ from each side, we get 3π‘₯>βˆ’1; and by dividing each side by 3, we get π‘₯>βˆ’13.
  3. 5π‘₯βˆ’3+3π‘₯βˆ’1>π‘₯+3; by adding the like terms, we get 8π‘₯βˆ’4>π‘₯+3; by subtracting π‘₯ from each side, we get 7π‘₯βˆ’4>3; by adding 4 to each side, we get 7π‘₯>7; and by dividing each side by 7, we get π‘₯>1.

By combining (1) and (3), we get 1<π‘₯<5. As we are told that π‘₯ is an integer, we find that π‘₯=2,3,4or.

Note that (2) did not give us any valuable information as π‘₯ is a length and is thus positive, and (3) gave that π‘₯>1.

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