In this explainer, we will learn how to define the triangle inequality and identify
whether the given side lengths are valid for constructing a triangle.
We can compare the sizes of any relatable quantities that we can represent with
numbers using inequalities. Using inequalities in this way allows us to solve
real-world problems where we need to compare the sizes or amounts of things. For
example, a company may wish to know the shortest distance between two places to
save on fuel and travel time.
This idea of the shortest distance combined with inequalities gives motivation to
an incredibly useful inequality known as the triangle inequality. To introduce this
concept, we start with the idea that the shortest distance between any two points
is a straight line. This is a fundamental result in geometry that we use in
everyday life even without thinking about it.
For example, if we want to get a drink from across the room, we will instinctively
travel in a straight line since this is the shortest distance. Another way of
thinking about this being the shortest distance is to say all other paths are
the same distance or longer. In particular, this means that any path between
two points that travels via another point must be at least as long as the direct
path. One way of visualizing this is to think of traveling between two places,
say London and Cairo. The shortest path is to travel directly in a straight
line, and it would take longer to travel via Moscow.
Ignoring the locations for the moment, we can think of this result entirely in
terms of the triangle. The shortest distance between the points is the straight
line of length and the other path of length must be longer since it is not
direct. Hence,
Since the above journey forms a triangle, we know that the inequality is strict
(i.e., is strictly less than ). However, not all choices of journeys form a
triangle in this way. For example, a straight line from London to Cairo travels
over Athens. If we instead traveled via Athens, our paths would look like the
following.
Both paths are direct straight lines, so both paths are equally as long. We have
These two situations showcase a useful result: any side in a triangle must be
shorter than the sum of the other two sides. If the length of a side is equal
to the sum of the other two sides, then the vertices must be colinear, meaning
that the three sides form a line segment rather than a triangle. This is called
the triangle inequality, and we can state it formally as follows.
Theorem: The Triangle Inequality
In any triangle , the sum of the lengths of any two sides is greater than the
length of the third side.
This means
We can use the triangle inequality to determine if it is possible to construct a
triangle with three given side lengths as we will see in our first example.
Example 1: Using the Triangle Inequality to Determine Whether a Triangle Is Possible
Is it possible to form a triangle with side lengths
6 m,
7 m, and
18 m?
Answer
We recall that the triangle inequality tells us that the sum of the lengths of any
two sides must be greater than the length of the third side. If we sum the lengths
of the two shortest lengths, we get
This is shorter than the third length, so these lengths do not satisfy the triangle
inequality. Hence, we cannot construct a triangle with these lengths.
Using the triangle inequality, we can show when it is not possible to construct a
triangle of given side lengths. However, we can also ask the reverse question: if
three side lengths satisfy the triangle inequality, will they always form a
triangle? As it turns out, the answer to this question is βyes,β and we can prove
it as follows.
Say we want to construct a triangle of side lengths , , and
and these values satisfy the triangle inequality. We can assume that the side of
length is the longest side. Letβs start by drawing a line segment of length
and labeling the endpoints of this line segment and
.
Suppose that the side of length is connected to point . If we consider all the
possible directions this side can be placed, we can see that they form a circle of
radius around , as shown below. We then trace a circle of radius
centered at
, where we note that the circle will intersect
since .
Letβs now do the same thing for the side of length
by tracing a circle of radius centered at
. Once again, we know that , so
this circle must intersect .
Since , the circles must overlap, as the sum of their radii
is greater than the distance between their centers. In fact, we know they overlap
on a point not on since the inequality is
strict. We then call a point of intersection between the circles ,
and we get the following.
We note that if we used the other intersection point of the circle, it would also
form a triangle with the same side lengths. We have shown the following result.
Theorem: Construction of Triangles That Satisfy the Triangle Inequality
If , , and are positive real
numbers that satisfy the triangle inequality, then there exists a triangle with side
lengths , , and .
In our next example, we will use the theorem to determine which of three given
choices are possible lengths for the sides of a triangle.
Example 2: Determining the Possible Lengths of a Triangleβs Sides
Which of the following lists could be the lengths of sides of a
triangle?
5, 2, 8
5, 3, 8
2, 5, 6
Answer
We begin by recalling that a triplet of positive numbers can only be the
side lengths of a triangle if they satisfy the triangle inequality that
states that the sum of the lengths of any two sides must be greater than
the length of the third side.
Let us consider the longest side in each case since it is the only side that
has the possibility of being greater than the two other sides.
In choice , we note that , so the sums of the sides of lengths 5 and
2 is not greater than the length of the remaining side. So, this triplet
does not satisfy the triangle inequality, and so they cannot be the side
lengths of a triangle.
In choice , we note that . Once again, the sum of the two shortest
sides is not greater than the third. This means that triplet does not
satisfy the triangle inequality, and so they cannot be the side lengths
of a triangle.
Finally, in choice , we see that
All three sums are larger than the length of the third side, so these 3
lengths satisfy the triangle inequality. It is worth noting that we
only needed to consider the first inequality since we just need to check
if the sum of the two shortest sides is greater than the longest side.
Hence, the triplet 2, 5, 6 can be the lengths of the sides of a triangle.
It is worth noting that we can construct the triangle in the previous example
by drawing a line of length 6 cm and then tracing arcs of circles of radii
5 cm
and 2 cm centered at the
end points of the line to get the following.
In our next example, we will find the range of all possible values for an
unknown in a missing side length of a triangle using the triangle
inequality.
Example 3: Finding the Range of All Possible Values of a
Triangleβs Side Length given the Lengths of the Other Sides
Find the range of all possible values of if
cm,
2 cm, and
25 cm
represent the lengths of the sides of a triangle.
Answer
We first recall that the side lengths in a triangle must be positive and
satisfy the triangle inequality. We also recall that if a triplet of three
numbers satisfies the triangle inequality, then these values can represent
the side lengths of a triangle.
The triangle inequality states that the sum of the lengths of any two sides
must be greater than the length of the third side. We can use this to
construct three inequalities that the side lengths must satisfy for the
values to represent the lengths of the sides of a triangle. It is worth
noting that since , we do not need to consider the inequality for 2
because it will always be smaller than 25 plus the other side length
(since it must be positive).
We have
We will solve both inequalities for . Simplifying the first inequality
gives
Subtracting 8 from both sides of the inequality yields
Simplifying the second inequality yields
Subtracting 6 from both sides of the inequality gives us
Therefore, we need and
. We can write this as the compound
inequality .
In our next example, we will combine the triangle inequality with our knowledge
of isosceles triangles to determine the missing side length in an isosceles
triangle.
Example 4: Finding the Missing Side Length of an Isosceles Triangle
If is an isosceles triangle with and
, find
.
Answer
We know that an isosceles triangle must have a pair of congruent sides. So,
for to be an isosceles triangle, the missing side
must be congruent to one of or
. This means the possible lengths of
are
2 cm and
5 cm.
We then recall that the side lengths in any triangle must satisfy the
triangle inequality. The triangle inequality tells us that the sum of the
lengths of any two sides must be greater than the length of the third side
and if this holds true, then the triangle can be constructed.
Therefore, we can check each of the triplets 2, 2, 5 and 2, 5, 5 to determine
if they can represent the side lengths of a triangle.
Consider the first triplet, 2, 2, 5. We see that
, so a side of length
5 would be longer than two sides of length 2, which means that this triplet
does not satisfy the triangle inequality.
For the second triplet, 2, 5, 5, we see that
It is worth noting that we only needed to consider the first inequality
since we just need to check if the sum of the two shortest sides is greater
than the longest side.
Hence, all three sums are larger than the length of the third side, which
means that this triplet satisfies the triangle inequality.
Therefore, .
In our final example, we will prove a relationship between the side lengths
of a triangle and its perimeter using the triangle inequality.
Example 5: Using the Triangle Inequality to Complete a Proof
Consider triangle , where
,
, and
.
Fill in the blanks using , ,
, , or = in the following
statements.
For any side of the triangle, say .
Therefore, we can conclude that .
If we denote the perimeter of the triangle by , then
.
Answer
Part 1
The triangle inequality tells us that the sum of the lengths of any two
sides must be greater than the length of the third side. This means that
the sum of the sides of lengths and
must be greater than . Giving us
Hence, the correct symbol is >.
Part 2
If we add to both sides of the inequality in part 1,
we get
Hence, the correct symbol is >.
Part 3
We recall that the perimeter of a polygon is equal to the sum of all of
its side lengths. So, in a triangle with sides of lengths
cm,
cm, and
cm, if the
perimeter is cm,
it must be
This is equal to the left-hand side of the previous inequality, so we have
Dividing both sides of the inequality by 2 gives us
Hence, the correct symbol is >.
In the previous example, we proved that half the perimeter of a triangle is
greater than any of the side lengths or, alternatively, that the perimeter of a
triangle is always greater than twice any of its side lengths.
Letβs finish by recapping some of the important points from this
explainer.
Key Points
The triangle inequality tells us that in any triangle , the sum of the lengths
of any two sides is greater than the length of the third side.
This means
If the length of one side of a triangle is equal to the sum of the lengths of the
other two sides, then the vertices are colinear, and so the shape is not really
a triangle but a line.
If , , and are positive
real numbers that satisfy the triangle inequality, then there exists a triangle
with side lengths , , and
.
We can prove some geometric relationships using the triangle inequality. For
example, the perimeter of any triangle is always greater than twice any of its
side lengths.
Join Nagwa Classes
Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!