Inequalities in One Triangle

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  • 0:04 Greater Angle/Greater…
  • 1:37 The Triangle…
  • 3:24 Example
  • 4:25 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Triangles show up all the time in the world around us. This lesson discusses some of the inequalities that describe the lengths of the sides of a triangle and the measurements of the angles. We'll look at theorems, proofs, and applications involving these inequalities.

Greater Angle/Greater Side Theorem

While planning a day trip to an amusement park, suppose you are looking at a map of the park, and you connect three different rides using straight lines, creating a triangle, ΔRST. A triangle is a three-sided and three-angled polygon. There are some interesting inequalities that relate the sides and angles of a triangle. These inequalities allow us to find various characteristics of a triangle.


ineqtri1


For example, consider a triangle, ΔABC, with side lengths a, b, and c opposite the corresponding angle, it will always be the case that a > b, if and only if ∠A > ∠B. In other words, longer sides always lie opposite larger angles. This is sometimes called the greater angle and greater side theorem.


ineqtri2


This theorem can be really useful when analyzing triangles. To illustrate this, suppose you measure the angles of your triangle on your map using a compass, and you end up with the following:

  • T = 72 degrees
  • S = 39 degrees
  • R = 69 degrees

Based on this and our greater angle and greater side theorem, can you determine which two sides have the greatest distance between them? Can you determine which two rides have the greatest distance between them? If you're thinking it's the Roller Coaster and the Sinking Ship, then you're correct! By our theorem, the longest side of the triangle will be opposite the greatest angle, which is ∠T. The side opposite ∠T is the line connecting the Roller Coaster and the Sinking Ship, so these two rides have the greatest distance between them.

Pretty neat, huh? Let's explore another very well-known theorem about inequalities in triangles called the triangle inequality theorem.

The Triangle Inequality Theorem

The triangle inequality theorem is a theorem that describes how the lengths of the sides of a triangle relate to one another. It states that in a triangle, the sum of any two sides of a triangle must be longer than the third side.


ineqtri3


The proof of this theorem relies on the shortest distance theorem, which states that the shortest distance between a point, x, and a line, l, is the line perpendicular to the line l, passing through x.

To prove the triangle inequality theorem, we would need to prove that all three inequalities are true. Let's consider just one of the inequalities, since the other two would be proved in the exact same manner.

First draw a triangle, ΔABC, and also draw a line segment from point B perpendicular to side AC, call the intersection point D. We'll prove that:

  • BC + AB > AC


ineqtri4


Now, by the shortest distance theorem, we have that CD is the shortest distance from C to BD, so CD < BC. Similarly, AD is the shortest distance from A to BD, so AD < AB. Thus, we have the following two inequalities:

  • CD < BC
  • AD < AB

If we add corresponding sides of each inequality, we get:

  • CD + AD < BC + AB

We're almost there! Notice that CD + AD = AC, so we plug in AC for CD + AD in the inequality to get:

  • AC < BC + AB or BC + AB > AC

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