In this video, we will learn how to define the triangle inequality and identify if the given side lengths are valid for constructing a triangle. We will discover that a triangle with side lengths 𝑎, 𝑏, and 𝑐 only exists if the two shorter side lengths have a sum that is greater than the third side length.
Let’s begin by considering how we might construct a triangle using a ruler and a pair of compasses. First, we will use a ruler to sketch the base of our triangle, naming it segment 𝐵𝐶 and giving it a length of 𝑎. Next, we will construct a circle with radius 𝑐 centered at point 𝐵. Then, we construct a second circle with radius 𝑏 centered at point 𝐶. Now, if we sketch a point 𝐴 at the intersection of circle 𝐵 and 𝐶, we will be able to form triangle 𝐴𝐵𝐶.
Now, it’s important to notice that when we constructed these two circles at the end points of segment 𝐵𝐶, the circles had two points of intersection. And we could’ve formed a triangle with vertex 𝐴 at either of these two points of intersection. At this point, it will be helpful for us to consider what we know about the lengths 𝑎, 𝑏, and 𝑐 for this first construction.
We recall that the shortest distance between two points is a straight line, so that means that the shortest distance between point 𝐵 and 𝐶 is length 𝑎. Therefore, any other path from point 𝐵 to point 𝐶 would be longer than the length 𝑎. So, let’s say that we made a detour from point 𝐵 to point 𝐴 before going to point 𝐶. That distance is represented by the sum of lengths 𝑏 plus 𝑐, which is certainly longer than length 𝑎. This conclusion leads us to wonder what if 𝑏 plus 𝑐 was not greater than 𝑎 but instead 𝑏 plus 𝑐 equaled 𝑎.
To explore this idea, we will erase our original construction and create a new one, where 𝑏 plus 𝑐 equals 𝑎. We will begin again with the segment 𝐵𝐶 of length 𝑎. But now, we’re working with a restriction that says 𝑏 plus 𝑐 must equal 𝑎. So after using our compass to construct circle 𝐵 with radius 𝑐 and circle 𝐶 with radius 𝑏, we see that these two circles meet at a point of tangency, and we’ll name that point 𝐴. In our first construction, point 𝐴 became the third vertex of a triangle. But now that 𝑏 plus 𝑐 equals 𝑎, we don’t have a triangle at all; we actually have a line segment.
Let’s stop to consider what we have learned so far from these first two constructions. We have learned that to form a triangle from lengths 𝑎, 𝑏, and 𝑐, 𝑏 plus 𝑐 can be greater than 𝑎, but 𝑏 plus 𝑐 cannot equal 𝑎.
Now, there’s only one more possibility to be considered: can 𝑏 plus 𝑐 be less than 𝑎? Let’s do a new construction to examine this possibility. As shown in this third construction, if the sum of the two radii is less than 𝑎, then these two circles will never intersect at all and they certainly will not form a triangle. Therefore, we conclude that to form a triangle with side lengths 𝑎, 𝑏, and 𝑐, it is perfectly fine for 𝑏 plus 𝑐 to be greater than 𝑎, as we saw in our first construction. However, it is not possible to form a triangle when 𝑏 plus 𝑐 equals 𝑎 or when 𝑏 plus 𝑐 is less than 𝑎.
Now, we will clear the screen in order to make space to write out the formal expression of this idea, called the triangle inequality theorem. The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This means that for any triangle 𝐴𝐵𝐶, 𝐴𝐵 plus 𝐴𝐶 is greater than 𝐵𝐶, 𝐴𝐵 plus 𝐵𝐶 is greater than 𝐴𝐶, and 𝐴𝐶 plus 𝐵𝐶 is greater than 𝐴𝐵. We can use the triangle inequality theorem to determine if it is possible to construct a triangle with three given side lengths. Let’s put this into practice in our first example.
Is it possible to form a triangle with side lengths six meters, seven meters, and 18 meters?
To answer this question, we will need to recall the triangle inequality theorem, which tells us that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This means that for any triangle 𝐴𝐵𝐶, 𝐴𝐵 plus 𝐴𝐶 is greater than 𝐵𝐶, 𝐴𝐵 plus 𝐵𝐶 is greater than 𝐴𝐶, and 𝐴𝐶 plus 𝐵𝐶 is greater than 𝐴𝐵.
We are being asked if it’s possible to form a triangle with the lengths of six meters, seven meters, and 18 meters. For this to be the case, these three values must make all three triangle inequalities true. So, we will be checking to see if 18 plus seven is greater than six, if 18 plus six is greater than seven, and if six plus seven is greater than 18.
The first inequality is true because 25 is greater than six. The second inequality is also true because 24 is greater than seven. But when we add the two shorter side lengths, we get 13, which is not greater than the third side length of 18. This means that we cannot form a triangle with side lengths six meters, seven meters, and 18 meters.
As we see in this problem, it is most important to check the sum of the two shorter side lengths. That is because this will be the inequality that is most likely to fail. So, it is helpful to note that we could check this inequality first in the future. And if that inequality fails, then we wouldn’t even need to go on to check the other two inequalities.
The following theorem answers the question “if three side lengths satisfy the triangle inequality, will they always form a triangle?” The answer to this question is yes, because according to the following theorem, if 𝑎, 𝑏, and 𝑐 are positive real numbers that satisfy the triangle inequality, then there exists a triangle with side lengths 𝑎, 𝑏, and 𝑐. This means that we can verify that positive real numbers, such as two, five, and six, do form a triangle, whereas the positive real numbers three, five, and eight do not form a triangle because the two shorter sides three and five do not add to a sum greater than eight.
In our next example, we will find the range of all possible values for an unknown side length of a triangle using the triangle inequality.
Find the range of all possible values of 𝑥 if 𝑥 plus six centimeters, two centimeters, and 25 centimeters represent the lengths of the sides of a triangle.
To find the range of all possible values of 𝑥, we must first recall that the side lengths in a triangle must be positive and satisfy the triangle inequality. If these three positive real numbers satisfy the triangle inequality, then these values can represent the side lengths of a triangle. The triangle inequality states that the sum of any two sides of a triangle must be greater than the length of the third side. We will use this fact to construct the three inequalities that the side lengths must satisfy in order to form a triangle.
First, we write out our three inequalities using every combination of 𝑥 plus six, two, and 25. Then, we will solve each inequality for 𝑥 to see what new information it will tell us about the possible range of values. By simplifying the left side of the first inequality, we have 𝑥 plus eight is greater than 25. Subtracting eight from both sides yields the strict inequality 𝑥 is greater than 17. This is certainly an important piece of information that tells us that the range of all possible values of 𝑥 must be above 17.
Solving the second inequality doesn’t give us any new information since we know that side lengths of a triangle must be positive and we already have that 𝑥 is greater than 17. Subtracting six from both sides of the last inequality gives us 21 is greater than 𝑥. If we turn this inequality around, it reads that 𝑥 is strictly less than 21. Since we need 𝑥 to be greater than 17 and less than 21, we can write our final answer as a compound inequality: 17 less than 𝑥 less than 21. We found this answer by constructing three inequalities according to the triangle inequality theorem with side lengths 𝑥 plus six, two, and 25.
In our last example, we will combine the triangle inequality theorem with our knowledge of isosceles triangles to determine the missing side length in an isosceles triangle.
If 𝐴𝐵𝐶 is an isosceles triangle with 𝐴𝐵 equal to two centimeters and 𝐵𝐶 equal to five centimeters, find 𝐴𝐶.
To begin, we will recall the definition of an isosceles triangle. An isosceles triangle is a triangle that has two congruent sides. The congruent sides are called the legs of the triangle, and the third side is called the base. The following triangle diagrams are not drawn to scale, but they can help us think through the possible arrangement of sides in this triangle. For triangle 𝐴𝐵𝐶 to be isosceles, the missing side 𝐴𝐶 must be congruent to either side 𝐴𝐵 or side 𝐵𝐶. If 𝐴𝐵 is congruent to 𝐴𝐶, then 𝐴𝐶 equals two. If 𝐵𝐶 is congruent to 𝐴𝐶, then 𝐴𝐶 has to equal five.
To determine whether the third side length is two centimeters or five centimeters, we will need to recall the triangle inequality theorem. The triangle inequality tells us that the sum of the lengths of any two sides of a triangle 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 two, two, five and two, five, five to determine if they can represent the side lengths of a triangle.
When considering the two, two, five triplet, we will need to check if the sum of the lengths of the two shorter sides is greater than the length of the third side. However, two plus two is less than five. This means that this triplet does not satisfy the triangle inequality. For the second triplet, we see that two plus five is greater than five, five plus two is greater than five, and five plus five is greater than two. Since all three sums are larger than the length of the third side, this means that the triplet satisfies the triangle inequality.
We have shown that the only possibilities for the length of 𝐴𝐶 were two centimeters or five centimeters. Then, we found out that a triangle with side lengths two centimeters, two centimeters, and five centimeters cannot exist because the sum of the two shorter side lengths is not greater than the third side length, whereas the lengths two centimeters, five centimeters, and five centimeters fulfills the requirements of all three triangle inequalities. Therefore, the third side of the isosceles triangle 𝐴𝐶 has a length of five centimeters.
We can now summarize some key points of this video. The central idea of this video was the triangle inequality, which tells us that in any triangle 𝐴𝐵𝐶 the sum of the lengths of any two sides must be greater than the length of the third side. That is, 𝐴𝐵 plus 𝐴𝐶 is greater than 𝐵𝐶, 𝐴𝐵 plus 𝐵𝐶 is greater than 𝐴𝐶, and 𝐴𝐶 plus 𝐵𝐶 is greater than 𝐴𝐵. We also saw that if 𝑎, 𝑏, and 𝑐 are positive real numbers that satisfy the triangle inequality, then there exists a triangle with side lengths 𝑎, 𝑏, and 𝑐.