### Video Transcript

In this video, weβll 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, itβs worth reminding ourselves of the Pythagorean
theorem and its converse.

The Pythagorean theorem states that
if triangle π΄π΅πΆ is a right-angled triangle at π΅, then the square of the side
length π΄πΆ is equal to π΄π΅ squared plus π΅πΆ squared. In other words, the square of the
longest side, thatβs 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. Thatβs 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
π΄π΅πΆ with its longest side opposite π΅, the square of the longest side π΄πΆ 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 π΅
is 90 degrees. And so the triangle is a right
triangle.

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

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

Letβs 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 πππ, ππ
squared is greater than ππ squared minus ππ squared. What type of angle is π?

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

Now weβve been given the
inequality ππ squared is greater than ππ squared minus ππ squared. And if we add ππ squared to
both sides to eliminate the negative term, we have ππ squared plus ππ
squared is greater than ππ squared. So on the right, weβve isolated
the side ππ, which is opposite the angle at π. Now rewriting our inequality so
we can compare it to the Pythagorean inequality, we have ππ squared is less
than the sum of the squares of the other two sides. This corresponds to the second
inequality. And so π 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 donβt 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 π΄π΅πΆ has side
lengths π΄π΅ equals seven centimeters, π΅πΆ equals nine centimeters, and π΄πΆ
equals 10 centimeters. Determine the angle with the
greatest measure in triangle π΄π΅πΆ. And determine the type of
triangle π΄π΅πΆ 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, thatβs the angle at
π΅, 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 π΄πΆ squared is 10 squared,
which is equal to 100. Now the sum of the squares of
the other two sides π΄π΅ and π΅πΆ is seven squared plus nine squared. Thatβs 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 π΅ must be an acute angle. Therefore, we have that π΅ is
the angle with the largest measure. And since π΅ is an acute angle,
triangle π΄π΅πΆ is an acute triangle.

Letβs 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
π΅πΆπ· in terms of its angles.

To find the type of triangle
π΅πΆπ·, 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, π΄πΆ, is greater than the sum of the squares of the
other two sides, then the angle opposite the longest side is obtuse. If itβs less than the sum of
the squares of the other two sides, the angle is acute. And if itβs equal to the sum of
squares, then the angle is a right angle.

We canβt apply this theorem yet
though, since we donβt know the lengths of sides π΅π· and πΆπ·. And to find these, we first
need to know what type of triangle triangle π΄π΅πΆ is. So letβs 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
π΄π΅πΆ is a right triangle with right angle at π΅, since π΅ is opposite the
hypotenuse.

Now since side π΅π· is a line
from the vertex π΅, which we can see bisects the opposite side, and π΅ is a
right angle, π΅π· is a median of triangle π΄π΅πΆ. Recalling then that the length
of a median at the right angle of a right angle triangle is half the hypotenuse,
we see that π΅π· is one half of 135. Thatβs 67.5 centimeters.

Noting also that since πΆπ· and
π·π΄ 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
π΅πΆπ·. To begin with though, we can
note that angle π΄πΆπ΅ is an angle in a right triangle, so itβs acute. Also, angle πΆπ΅π· is smaller
than the right angle π΄π΅πΆ, so πΆπ΅π· is also acute. Or alternatively, we could note
that since triangle πΆπ·π΅ is isosceles, angles πΆ and π΅ are the same and
therefore must be acute.

Finally, for angle πΆπ·π΅, we
compare the square of the side length opposite, thatβs the longest side πΆπ΅, 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 πΆπ΅ squared, thatβs 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 πΆπ·π΅ must be an acute angle. Hence, since the angle with the
greatest measure in triangle π΅πΆπ· is an acute angle, π΅πΆπ· is an acute
triangle.

Letβs 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.

π΄π΅πΆπ· is a
parallelogram. If π΄πΆ equals 13 centimeters,
π΄π· equals 13 centimeters, and π·πΆ equals five centimeters, what is the type
of triangle π΄π·πΆ?

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

Taking the angle at π· 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 π΄πΆ
squared, that is 13 squared, equals 169 and that π΄π· squared plus π·πΆ squared
equals 13 squared plus five squared. And thatβs equal to 194. Hence, π΄πΆ squared is less
than π΄π· squared plus π·πΆ squared. And so angle πΆπ·π΄ is an acute
angle. Angle π΄πΆπ· is the same, so
this is also acute. And since these angles have the
largest measure in triangle π΄π·πΆ, angle πΆπ΄π· must be smaller than them. Hence, the third angle, angle
πΆπ΄π· is also acute. Since all three angles are
acute and, in particular, the angle with the largest measure is acute, triangle
π΄π·πΆ is an acute triangle.

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

Classify the triangle πΈπΉπΆ,
where side length π΅πΉ equals root three centimeters and side length π΄πΈ equals
root six centimeters and where π΄π΅πΆπ· is a rectangle.

To determine the type of
triangle πΈπΉπΆ, 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 donβt yet
know the lengths of the sides of our triangle πΈπΉπΆ. But we do know that itβs
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
πΉπ΄πΈ, since π΄πΉ is equal to πΉπ΅ , thatβs root three, we can find the length
of side πΈπΉ. πΈπΉ squared equals root six
squared plus root three squared, which is nine. And taking the positive square
root on both sides β positive since weβre looking for the length β we have side
length πΈπΉ equal to three centimeters.

Now if we consider side length
πΈπΆ, we know that π·πΈ equals πΈπ΄, which is root six centimeters, and that
π·πΆ equals two root three, since itβs the same length as side π΄π΅. So we have πΆπΈ squared equals
πΈπ· squared plus π·πΆ squared. Thatβs root six squared plus
two root three squared, which is 18. And taking the positive square
root gives πΆπΈ equal to three root two centimeters.

Using the same method for side
πΆπΉ, we have πΆπΉ squared equals πΆπ΅ squared plus π΅πΉ squared. πΆπΉ squared is therefore
27. And so πΆπΉ is equal to three
root three. We now have all three side
lengths for the triangle πΈπΉπΆ. Now since three root three is
greater than three root two which is greater than three, our longest side is
πΆπΉ.

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 πΆπΉ. Thatβs angle πΆπΈπΉ. We have that πΆπΉ squared
equals three root three squared, and thatβs 27. Next, we have the sum of the
squares of the other two sides, πΈπΉ squared plus πΆπΈ squared, which equals
three squared plus three root two squared. And thatβs nine plus 18, which
is also equal to 27. Hence, the square of the
longest side in triangle πΈπΉπΆ 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 πΆπΈπΉ is a right angle.

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

Letβs complete this video by
reminding ourselves of some of the key points weβve 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, itβs called a right triangle.

We know that the largest angle in
any triangle is opposite the triangleβs longest side, which leads us to the
Pythagorean inequality theorem, where if the angle at π΅ 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 π΅ 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.