Lesson Video: Pythagorean Inequality Theorem Mathematics • 8th Grade

Video Transcript

In this video, we’ll learn how to determine whether an angle in a triangle is acute, right, or obtuse by using the Pythagorean inequality. But before we begin discussing the Pythagorean inequality theorem, it’s worth reminding ourselves of the Pythagorean theorem and its converse.

The Pythagorean theorem states that if triangle 𝐴𝐵𝐶 is a right-angled triangle at 𝐵, then the square of the side length 𝐴𝐶 is equal to 𝐴𝐵 squared plus 𝐵𝐶 squared. In other words, the square of the longest side, that’s the side opposite the right angle, equals the sum of the squares of the other two sides.

The converse of the Pythagorean theorem, on the other hand, tells us that if the square of the length of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle. That is, the angle opposite to the longest side is a right angle. If we know the side lengths of a triangle, we can use this result to prove whether or not a given triangle is a right triangle.

For example, suppose we know the side lengths of a triangle to be six, eight, and 10 centimeters. We know that 10 squared equals 100, eight squared is 64, and six squared is 36. And since 64 plus 36 is equal to 100, by the converse of the Pythagorean theorem, the angle opposite the longest side must be a right angle and the triangle is a right triangle.

Similarly, suppose we have a triangle with side lengths two, three, and four, then since four squared is 16, two squared is four, and three squared is nine, we find that two squared plus three squared equals 13, which does not equal 16. Hence, the triangle with these side lengths is not a right triangle.

We can extend these ideas further with respect to non-right triangles, by first recalling the definitions of obtuse and acute triangles. Recall that an obtuse triangle has an obtuse internal angle. That’s an internal angle greater than 90 degrees. And in an acute triangle, all internal angles are acute, that is, less than 90 degrees.

We can determine whether a triangle is acute, right, or obtuse by extending the Pythagorean theorem to the Pythagorean inequality theorem. This tells us that for a triangle 𝐴𝐵𝐶 with its longest side opposite 𝐵, the square of the longest side 𝐴𝐶 is greater than the sum of the squares of the other two sides. Also, if the square of the longest side is less than the sum of the squares of the other two sides, then the triangle is an acute triangle. And of course, in between these two, we have the converse of the Pythagorean theorem such that if the square of the longest side equals the sum of the squares of the other two sides, then the angle 𝐵 is 90 degrees. And so the triangle is a right triangle.

To see why the inequality theorem works, consider two fixed length sides 𝐴𝐵 and 𝐵𝐶 connected by a hinge at 𝐵. So we keep 𝐴 and 𝐵 fixed but rotate point 𝐶 to construct different triangles. We can rotate these fixed-length sides into different legs of a triangle. And we’re interested in what happens to the side lengths 𝐴𝐶 one, 𝐴𝐶 two, and 𝐴𝐶 three. Rotating the side to make the angle at the hinge 𝐵 a right angle gives us a right triangle. And so we know that 𝐴𝐶 two squared equals 𝐴𝐵 squared plus 𝐵𝐶 two squared. And that’s the Pythagorean theorem.

Increasing the measure of angle 𝐵 will make the side length 𝐴𝐶 longer, as in the first diagram. And decreasing the measure of angle 𝐵 will make 𝐴𝐶 shorter, as in the third diagram. Hence, if angle 𝐵 is obtuse, then the square of side length 𝐴𝐶 is greater than the sum of the squares of the other two sides. And if angle 𝐵 is acute, the square of 𝐴𝐶 is less than the sum of the squares of the other two sides. Thus, we have the Pythagorean inequality theorem.

Let’s look now at some examples of how we can apply this inequality theorem to determine the type of angle in a triangle given information about its side lengths.

In triangle 𝑋𝑌𝑍, 𝑌𝑍 squared is greater than 𝑋𝑍 squared minus 𝑋𝑌 squared. What type of angle is 𝑌?

To determine what type of angle 𝑌 is in triangle 𝑋𝑌𝑍, we can apply the Pythagorean inequality theorem. This tells us that if 𝑋𝑌𝑍 is a triangle with its longest side opposite 𝑌, then if the square of the side opposite 𝑌 is greater than the sum of the squares of the other two sides, angle 𝑌 is obtuse. Alternatively, if the square of the side opposite angle 𝑌 is less than the sum of the squares of the other two sides, then 𝑌 is acute. And of course, we recall the Pythagorean theorem, or actually its converse, that says if the square of the side opposite 𝑌 is equal to the sum of the squares of the other two sides, 𝑌 is a right angle.

Now we’ve been given the inequality 𝑌𝑍 squared is greater than 𝑋𝑍 squared minus 𝑋𝑌 squared. And if we add 𝑋𝑌 squared to both sides to eliminate the negative term, we have 𝑌𝑍 squared plus 𝑋𝑌 squared is greater than 𝑋𝑍 squared. So on the right, we’ve isolated the side 𝑋𝑍, which is opposite the angle at 𝑌. Now rewriting our inequality so we can compare it to the Pythagorean inequality, we have 𝑋𝑍 squared is less than the sum of the squares of the other two sides. This corresponds to the second inequality. And so 𝑌 is an acute angle.

Now to determine what type a triangle is from its side lengths, we could check every angle using the Pythagorean inequality theorem. But we don’t really need to do this if we recall the triangle property that the angle with the largest measure in a triangle is always opposite the longest side. In our next example, we use this property to find the largest angle in a triangle from its side lengths and then the Pythagorean inequality theorem to find the type of triangle.

The triangle 𝐴𝐵𝐶 has side lengths 𝐴𝐵 equals seven centimeters, 𝐵𝐶 equals nine centimeters, and 𝐴𝐶 equals 10 centimeters. Determine the angle with the greatest measure in triangle 𝐴𝐵𝐶. And determine the type of triangle 𝐴𝐵𝐶 in terms of its angles.

To answer the first part, we recall that in a triangle, the angle with the greatest measure is always opposite the longest side. Hence, since the longest side in our triangle is 10 centimeters, the angle opposite this, that’s the angle at 𝐵, must have the largest measure.

Now for the second part of the question, to determine the type of triangle we have in terms of its angles, we can apply the Pythagorean inequality theorem, which tells us three things. First, that if the square of the longest side length in a triangle is greater than the sum of the squares of the other two sides, then the angle opposite the longest side is an obtuse angle. Second, if the square of the longest side is less than the sum of the squares of the other two sides, then the angle opposite is acute. And third, if the square of the longest side equals the sum of the squares of the other two sides, then the angle opposite is a right angle. This third part is actually the converse of the Pythagorean theorem.

Applying this to our triangle, where the longest side is 10 centimeters, we have 𝐴𝐶 squared is 10 squared, which is equal to 100. Now the sum of the squares of the other two sides 𝐴𝐵 and 𝐵𝐶 is seven squared plus nine squared. That’s 49 plus 81, which is equal to 130. We see then that the square of the longest side, which is 100, is less than the sum of the squares of the other two sides, which is 130. So by the Pythagorean inequality theorem, the angle 𝐵 must be an acute angle. Therefore, we have that 𝐵 is the angle with the largest measure. And since 𝐵 is an acute angle, triangle 𝐴𝐵𝐶 is an acute triangle.

Let’s look at an example now where we apply the Pythagorean inequality theorem along with geometric results to determine the type of triangle.

Determine the type of triangle 𝐵𝐶𝐷 in terms of its angles.

To find the type of triangle 𝐵𝐶𝐷, we begin by recalling that if we know the side lengths of a triangle, we can use the Pythagorean inequality theorem to determine its type. This tells us that if the square of the longest side, 𝐴𝐶, is greater than the sum of the squares of the other two sides, then the angle opposite the longest side is obtuse. If it’s less than the sum of the squares of the other two sides, the angle is acute. And if it’s equal to the sum of squares, then the angle is a right angle.

We can’t apply this theorem yet though, since we don’t know the lengths of sides 𝐵𝐷 and 𝐶𝐷. And to find these, we first need to know what type of triangle triangle 𝐴𝐵𝐶 is. So let’s use its side lengths to work this out. We have 135 squared equals 18,225, which is also equal to 81 squared plus 108 squared. And this means that triangle 𝐴𝐵𝐶 is a right triangle with right angle at 𝐵, since 𝐵 is opposite the hypotenuse.

Now since side 𝐵𝐷 is a line from the vertex 𝐵, which we can see bisects the opposite side, and 𝐵 is a right angle, 𝐵𝐷 is a median of triangle 𝐴𝐵𝐶. Recalling then that the length of a median at the right angle of a right angle triangle is half the hypotenuse, we see that 𝐵𝐷 is one half of 135. That’s 67.5 centimeters.

Noting also that since 𝐶𝐷 and 𝐷𝐴 are equal, they too must be half the length of the hypotenuse. And we can use the Pythagorean inequality theorem to determine the type of the largest angle in triangle 𝐵𝐶𝐷. To begin with though, we can note that angle 𝐴𝐶𝐵 is an angle in a right triangle, so it’s acute. Also, angle 𝐶𝐵𝐷 is smaller than the right angle 𝐴𝐵𝐶, so 𝐶𝐵𝐷 is also acute. Or alternatively, we could note that since triangle 𝐶𝐷𝐵 is isosceles, angles 𝐶 and 𝐵 are the same and therefore must be acute.

Finally, for angle 𝐶𝐷𝐵, we compare the square of the side length opposite, that’s the longest side 𝐶𝐵, to the sum of the squares of the other two sides. We have 81 squared equals 6,561. And 67.5 squared plus 67.5 squared equals 9,112.5. And so 𝐶𝐵 squared, that’s the square of the longest side, is less than the sum of the squares of the other two sides. By the Pythagorean inequality theorem then, angle 𝐶𝐷𝐵 must be an acute angle. Hence, since the angle with the greatest measure in triangle 𝐵𝐶𝐷 is an acute angle, 𝐵𝐶𝐷 is an acute triangle.

Let’s look at another example of how we can apply the Pythagorean inequality theorem to finding the type of a triangle, this time in a parallelogram.

𝐴𝐵𝐶𝐷 is a parallelogram. If 𝐴𝐶 equals 13 centimeters, 𝐴𝐷 equals 13 centimeters, and 𝐷𝐶 equals five centimeters, what is the type of triangle 𝐴𝐷𝐶?

We see that triangle 𝐴𝐷𝐶 is an isosceles triangle. And recalling that the angle in a triangle with the greatest measure is opposite the longest side, in triangle 𝐴𝐷𝐶 the angles at 𝐶 and 𝐷, which are equal, will have the largest measure. Choosing either one of the angles at 𝐶 and 𝐷, we can use the Pythagorean inequality theorem to confirm that these angles are acute.

Taking the angle at 𝐷 to work on, this theorem tells us three things. First, that if the square of the longest side is greater than the sum of the squares of the other two sides, then the angle opposite the longest side is an obtuse angle. Second, if the square of the longest side is less than the sum of squares of the other two sides, the angle is acute. And third, if the square of the longest side is equal to the sum of the squares of the other two, then the angle opposite is a right angle.

In our case, we have 𝐴𝐶 squared, that is 13 squared, equals 169 and that 𝐴𝐷 squared plus 𝐷𝐶 squared equals 13 squared plus five squared. And that’s equal to 194. Hence, 𝐴𝐶 squared is less than 𝐴𝐷 squared plus 𝐷𝐶 squared. And so angle 𝐶𝐷𝐴 is an acute angle. Angle 𝐴𝐶𝐷 is the same, so this is also acute. And since these angles have the largest measure in triangle 𝐴𝐷𝐶, angle 𝐶𝐴𝐷 must be smaller than them. Hence, the third angle, angle 𝐶𝐴𝐷 is also acute. Since all three angles are acute and, in particular, the angle with the largest measure is acute, triangle 𝐴𝐷𝐶 is an acute triangle.

In our final example, we use the Pythagorean inequality to classify a triangle inside a rectangle.

Classify the triangle 𝐸𝐹𝐶, where side length 𝐵𝐹 equals root three centimeters and side length 𝐴𝐸 equals root six centimeters and where 𝐴𝐵𝐶𝐷 is a rectangle.

To determine the type of triangle 𝐸𝐹𝐶, we can use the Pythagorean inequality theorem. This tells us that depending on whether the square of the longest side is greater than, less than, or equal to the sum of the squares of the other two sides, the angle opposite the longest side, and therefore the triangle itself, is either obtuse, acute, or right angled, respectively.

Now in our case, we don’t yet know the lengths of the sides of our triangle 𝐸𝐹𝐶. But we do know that it’s inscribed in a rectangle and that the corners of a rectangle are right angles. So we can use the Pythagorean theorem for right-angled triangles to find our three missing side lengths. If we start with triangle 𝐹𝐴𝐸, since 𝐴𝐹 is equal to 𝐹𝐵 , that’s root three, we can find the length of side 𝐸𝐹. 𝐸𝐹 squared equals root six squared plus root three squared, which is nine. And taking the positive square root on both sides — positive since we’re looking for the length — we have side length 𝐸𝐹 equal to three centimeters.

Now if we consider side length 𝐸𝐶, we know that 𝐷𝐸 equals 𝐸𝐴, which is root six centimeters, and that 𝐷𝐶 equals two root three, since it’s the same length as side 𝐴𝐵. So we have 𝐶𝐸 squared equals 𝐸𝐷 squared plus 𝐷𝐶 squared. That’s root six squared plus two root three squared, which is 18. And taking the positive square root gives 𝐶𝐸 equal to three root two centimeters.

Using the same method for side 𝐶𝐹, we have 𝐶𝐹 squared equals 𝐶𝐵 squared plus 𝐵𝐹 squared. 𝐶𝐹 squared is therefore 27. And so 𝐶𝐹 is equal to three root three. We now have all three side lengths for the triangle 𝐸𝐹𝐶. Now since three root three is greater than three root two which is greater than three, our longest side is 𝐶𝐹.

And recalling that the angle with the largest measure in a triangle is always opposite the longest side, we can apply the Pythagorean inequality theorem to determine the type of the angle opposite side 𝐶𝐹. That’s angle 𝐶𝐸𝐹. We have that 𝐶𝐹 squared equals three root three squared, and that’s 27. Next, we have the sum of the squares of the other two sides, 𝐸𝐹 squared plus 𝐶𝐸 squared, which equals three squared plus three root two squared. And that’s nine plus 18, which is also equal to 27. Hence, the square of the longest side in triangle 𝐸𝐹𝐶 is equal to the sum of the squares of the other two sides. And so by the third part of the Pythagorean inequality theorem, angle 𝐶𝐸𝐹 is a right angle.

Now since angle 𝐶𝐸𝐹 is a right angle and the angles of a triangle sum to 180 degrees, the other two angles, 𝐸𝐶𝐹 and 𝐸𝐹𝐶, must both be acute angles. Hence, as the angle with the largest measure in triangle 𝐸𝐹𝐶 is a right angle, triangle 𝐸𝐹𝐶 is a right triangle.

Let’s complete this video by reminding ourselves of some of the key points we’ve covered.

First, if all the angles in a triangle are acute, then the triangle is called an acute triangle. Next, if a triangle has an internal obtuse angle, then we call it an obtuse triangle. And third, if a triangle has an internal right angle, it’s called a right triangle.

We know that the largest angle in any triangle is opposite the triangle’s longest side, which leads us to the Pythagorean inequality theorem, where if the angle at 𝐵 is the angle with the largest measure, then if the square of the longest side is greater than the sum of the squares of the other two sides, angle 𝐵 is obtuse and the triangle is an obtuse triangle. If the square of the longest side is smaller than the sum of the squares of the other two sides, then the angle with the greatest measure is acute and the triangle is an acute triangle. And finally, if the square of the longest side and the sum of the squares of the other two sides are equal, the angle opposite is a right angle and the triangle is a right triangle.