Video Transcript
A triangle has sides of lengths five centimeters, eight centimeters, and 𝑥
centimeters. State the range of values that 𝑥 can take.
In this question, we are given the side lengths of a triangle as five and eight
centimeters. We need to use this to determine the range of possible values for the unknown
side. To answer this question, we need to recall two things. First, we can recall that the triangle inequality tells us that in any triangle the
sum of the lengths of any two of its sides must be larger than the length of the
third side. So, if a triangle has lengths 𝑎, 𝑏, and 𝑐, then 𝑎 plus 𝑏 must be greater than
𝑐.
We can use the triangle inequality to construct three inequalities that must be
satisfied by the lengths of this triangle. We know that the sum of any two of its side lengths must be greater than the length
of the remaining side. So, we have that five plus eight must be greater than 𝑥, five plus 𝑥 must be
greater than eight, and eight plus 𝑥 must be greater than five. It is worth noting that 𝑥 is the length of a side in a triangle, so we know that 𝑥
is positive. This means that the final inequality will be satisfied regardless of the value of
𝑥.
The second property that we need to use is that if we have any three lengths — that
is, positive real numbers that satisfy the triangle inequality so the sum of any
pair of these lengths is greater than the third — then there is a triangle that has
these lengths as its side lengths. We can think of this as the reverse of the triangle inequality. The triangle inequality lets us check if a triangle is possible using its side
lengths. And this result lets us show that there is a triangle with the lengths as its side
lengths. This means that if we chose a value of 𝑥 that satisfies all three of these
inequalities, then that is the range of values that 𝑥 can take. Of course, we have already shown that the third inequality is satisfied provided 𝑥
is positive.
Let’s simplify each of the first two inequalities. In the first inequality, we have five plus eight is 13. We can rewrite this inequality as 𝑥 is less than 13. In the second inequality, we can subtract five from both sides of the inequality to
obtain that 𝑥 is greater than three. We can rewrite this as three is less than 𝑥. We can then combine these two inequalities into the single compound inequality three
is less than 𝑥 is less than 13.
We have shown that if 𝑥 is in this range, then the three values satisfy the triangle
inequality, so a triangle with these side lengths exists, and that 𝑥 must be in
this range for such a triangle to exist.
