Math Review of the Triangle Inequality

Overview:

Triangles have many different properties.  For example, the sum of all three angles is always equal to 180o. 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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