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Outside the Box Geometry

# Triangle Inequality

True or False?

The three sides of a triangle can be a set of any three lengths.

You’re making a triangle out of sticks and the first two sides $$(a$$ and $$b)$$ are 4 cm and 8 cm, respectively.

What are the possible lengths for the third side of the triangle, $$c\, ?$$

The triangle inequality theorem outlines what the constraints are on that third side:

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

In other words, once two side lengths have been chosen for two sides of a triangle, there are restrictions on how long that third side can be. We know that the third side has to be long enough to be able to connect the first two in the shape of a triangle, but short enough that the other two sides don't wind up end to end in a line.

For example, the side lengths of 4, 6, and 7 create a triangle while the side lengths of 4, 6, and 10 do not.

Given a triangle with one side length of 3, how many of these pairs of side lengths could complete the triangle?

A. 8 and 10

B. 8 and 11

C. 10,000 and 10,002

D. 100,000 and 100,002

E. 1,000,000 and 1,000,002

What is the length of the blue segment in the figure, given that it is a positive integer?

Triangle Y is an obtuse triangle. The side opposite the obtuse angle is 10 cm long and a second side is 8 cm long.

Carl claims that the $$3^\text{rd}$$ side might be 16 cm long because $$16 < 10 + 8.$$ Is Carl correct?

Triangle X is a right triangle. One of its sides is 4 cm long and another is 13 cm long. How many different lengths could the final side be?

There are many powerful ways in which knowing some properties of a triangle can give you partial or complete information about other properties. Here are some of the most fundamental triangle relationships:

1) The sum of the three internal angles in a triangle is always exactly $$180^\circ.$$

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

3) The longest side of a triangle will always be across from the largest internal angle.

4) Right triangles have exactly one $$90^\circ$$ angle. In a right triangle, the square of the length of the side opposite the $$90^\circ$$ angle equals the sum of the squares of the lengths of the other two sides. This property is generally known as the Pythagorean theorem.

The next several quizzes in this chapter will continue to explore what we can deduce from different kinds of partial information about a triangle.

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