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.
