# Math Review of the Triangle Inequality

**Overview:**

Triangles have many different properties. For example, the sum of all three angles is always equal to 180^{o}. The area of a triangle is equal to 1/2 times the base times the height. The triangle inequality property is another property that can be used in real-world situations.

**Straight Lines and Triangles**

Suppose A, B, and C are points that lie on a line. Then, the measurement of segment AB + the measurement of segment BC will equal the measurement of segment AC exactly. If segment AB = x and segment BC =y, segment AC will equal x +y. If points A, B, and C do not lie on the same line, the points will be the vertices of a triangle, and AB + BC will not equal AC.

**The Triangle Inequality**

According to the triangle inequality principle, the sum of the length of any two sides of the triangle is greater than the third side. That means that for any triangle ∆ABC, AB + BC ≥AC, AC + BC ≥AB , and AC + AB ≥BC. Suppose the length of AB =x, the length of BC =y, and the length of AC = z. Then x +y >z, x +z >y, and y +z >x.

**Applying the Principle**

Suppose side AB (or x) measures 15 units and side BC (or y) measures 22 units. That would leave 3 possible values. Either x +y >z , x +z >y, or y+z >x. Substituting the numbers for x and y, 15 +22 >z or 37>z. In the second inequality, 15 +z >22. If z > 22-15, z>7. The third inequality says that 22 + z> 15, which would make z>-7. The most meaningful statement is that z is between 7 and 37.

**How Is It Useful?**

Sometimes it is enough to know that a value is between a range of numbers. A door on a hinge forms a triangle that has a minimum value and a maximum value. If that door is to be installed within a small space, it would be important to know that it could open and close freely. Suppose that a freezer has to fit in a pantry that has a clearance of 40 inches. If the hinge forms a triangle with the smallest value of 7 inches when it is nearly closed, and a maximum value when it is fully open of 37 inches, the freezer would fit in the pantry. If the pantry had a clearance of 30 inches, it would not fit, because the maximum value of 37 inches would be too large for the space.

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