In this explainer, we will learn how to use the converse of the Pythagorean theorem to determine whether a triangle is a right
triangle.
We should already have learned about the Pythagorean theorem.
Theorem: The Pythagorean Theorem
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares
of the two shorter sides (called the legs).
Write for the length of the hypotenuse and and for the lengths
of the legs. The Pythagorean theorem can be expressed algebraically as
In view of this, it makes sense to ask the following question: Does this mean that a triangle with side lengths satisfying
the equation is necessarily a right triangle?
Let us assume that has side lengths , , and
and satisfies . Then, let be a
right triangle with side lengths , , and and a right angle at
.
By applying the Pythagorean theorem to , we find
Now, we know that . Furthermore, for the given and ,
the sum has a single value. Therefore, we must have . Since
and are lengths, we can take positive square roots of both sides of the last equation,
from which it follows that .
Since a triangle is defined uniquely by the lengths of its three sides, we conclude that
and are congruent. Consequently, must have a right
angle at .
This is a proof of the converse of the Pythagorean theorem.
Theorem: The Converse of the Pythagorean Theorem
If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is
a right triangle.
If , then the triangle has a right angle at .
A corollary of this result is that if , the triangle does not have a right angle at
.
Thus, the converse of the Pythagorean theorem allows us to determine whether a given triangle has a right angle.
How To: Determining Whether a Given Triangle Has a Right Angle
To find whether a given triangle has a right angle, we can do the following:
Identify the longest side of the triangle.
Calculate the square of the longest side and the sum of the squares of the other two side lengths.
If the two quantities are equal, the triangle has a right angle. If not, the triangle has no right angle.
Let us start with an example to check our understanding of the converse of the Pythagorean theorem.
Example 1: The Converse of the Pythagorean Theorem
What can the converse of the Pythagorean theorem be used for?
Demonstrating that a triangle is equilateral
Demonstrating that a triangle has a right angle
Finding the angles in a triangle
Finding lengths in an equilateral triangle
Demonstrating that a triangle is an isosceles triangle
Answer
First, recall the Pythagorean theorem, which states that in any right triangle, the square of the hypotenuse is equal to the
sum of the squares of the two shorter sides (called the legs). Writing for the length of the hypotenuse
and and for the lengths of the legs, we can express this theorem algebraically as
.
Note that the Pythagorean theorem applies to right triangles only; it shows that the presence of a right angle in a triangle
implies the relationship between its side lengths.
The converse of the Pythagorean theorem works in the opposite direction. It starts from a relationship between the side lengths
of a triangle and uses this to prove that the triangle has a right angle. In other words, write for the
length of the longest side of a triangle and and for the lengths of the two shorter
sides; the converse of the Pythagorean theorem tells us that if , then the triangle has a
right angle.
Reviewing the five possible answer choices, we conclude that the correct one is B: the converse of the Pythagorean theorem
can be used for demonstrating that a triangle has a right angle.
Next, we will use the converse of the Pythagorean theorem to test whether a given triangle is a right triangle.
Example 2: Identifying the Type of a Triangle by Applying the Converse of the Pythagorean Theorem
Is this triangle a right triangle?
Yes
No
Answer
First, we recall the converse of the Pythagorean theorem. Write for the length of the longest side of
a triangle and and for the lengths of the two shorter sides. The converse of the
Pythagorean theorem tells us that if , then the triangle has a right angle.
A corollary of this result is that if , the triangle does not have a right angle.
From the diagram, the given triangle has side lengths of 13 cm,
12 cm, and 5 cm,
so the longest side has length . Squaring this value gives
The two shorter sides have lengths and
. The sum of the squares of these side lengths is
Both calculations give the same answer, so and the converse of the Pythagorean theorem
implies this is a right triangle. Therefore, the correct answer is βYes,β which is A.
We will now look at an example without a diagram.
Example 3: Identifying the Type of a Triangle by Applying the Converse of the Pythagorean Theorem
Can the lengths 7.9 cm,
8.1 cm, and
5.3 cm form a right triangle?
No
Yes
Answer
Write for the length of the longest side of a triangle and and
for the lengths of the two shorter sides. The converse of the Pythagorean theorem tells us that if , then the triangle has a right angle. A corollary of this result is that if , then the triangle does not have a right angle.
We start by identifying the longest side length of the triangle, which is . Squaring this value gives
The shorter two sides have lengths
and . The sum of the squares of these
side lengths is
Comparing these values, clearly , so . We conclude
that the given lengths cannot form a right triangle, so the correct answer is βNo,β which is A.
The converse of the Pythagorean theorem can also be applied to solve geometric problems of greater complexity, sometimes in
conjunction with the Pythagorean theorem itself.
Example 4: Applying the Pythagorean Theorem and Its Converse to Determine Whether an Angle Is Right
Is a right triangle at ?
No
Yes
Answer
In this question, we are given two of the three side lengths of , namely,
and , and need to work out if it is a right triangle at .
From the diagram, we also know that is a right triangle at
and are given and , the lengths of its two shorter sides. Furthermore, the
missing side length of is , which is exactly half of
, the missing length of the hypotenuse of .
Therefore, as is a right triangle, we can apply the Pythagorean theorem to find
the length of its hypotenuse. Once we have this answer, we can halve it to provide the missing side length of
. Finally, we can apply the converse of the Pythagorean theorem in
to check whether it is a right triangle at .
Starting with , we write for , the length
of the hypotenuse. The Pythagorean theorem states that in any right triangle, the square of the hypotenuse is equal to the
sum of the squares of the two shorter sides. Hence, we can substitute the known values into the equation given by the
Pythagorean theorem, , as follows:
As is a length, it is positive, so
Next, we calculate the missing side length of by halving this value to get
. We now have all three side
lengths of : 19.1 cm,
15.6 cm, and
10.25 cm.
Writing for the length of the longest side of a triangle and and
for the lengths of the two shorter sides, we recall the converse of the Pythagorean theorem, which tells us that if
, then the triangle has a right angle. A corollary of this result is that if
, then the triangle does not have a right angle.
The longest side of has length , and squaring this value gives
The other two side lengths are and
. The sum of the squares of these
lengths is
Comparing these values, clearly , so .
By the converse of the Pythagorean theorem, we deduce that does not have a right
angle at . This means the correct answer is βNo,β which is A.
In our final example, we show how we can apply the converse of the Pythagorean theorem to help us find an area.
Example 5: Applying the Converse of the Pythagorean Theorem to Find the Area of a Triangle
A triangle has sides of lengths 36.4, 27.3, and 45.5. What is its area?
Answer
Here, we are given the three side lengths of a triangle. Writing for the length of the longest side and
and for the lengths of the two shorter sides, we recall the converse of the Pythagorean
theorem, which tells us that if , then the triangle has a right angle.
If we can prove that the given triangle is a right triangle, it will be straightforward to work out its area, for the following
reason. Remember that the legs of a right triangle form a right angle. If the lengths of the legs are and
, then would represent the area of a rectangle with side lengths
and . Therefore, the quantity , which is half of this area, represents the
area of the corresponding right triangle. This is shown in the diagram below.
We start by identifying the longest side of the triangle, which is . Squaring this value gives
The other two side lengths are and , and the sum of the squares of these
lengths is
Both calculations give the same answer, so , and the converse of the Pythagorean theorem
implies this is a right triangle.
Lastly, we can calculate the area of this right triangle. The two shorter sides (the legs) must form a right angle, and they
have lengths 36.4 and 27.3. We, therefore, obtain the area by multiplying these values and
then dividing the result by 2:
Thus, we have found that the area of the triangle is 496.86 square units.
Let us finish by recapping some key concepts from this explainer.
key Points
The converse of the Pythagorean theorem states that if the square of the longest side of a triangle is equal to the sum
of the squares of the other two sides, then the triangle is a right triangle.
If , then the triangle has a right angle at .
A corollary of this result is that if , then the triangle does not have a right angle
at .
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