3.8: Pythagorean Theorem, Part 3: Converse of the Pythagorean Theorem
Learning Objectives
- Understand the converse of the Pythagorean Theorem.
- Identify acute triangles from side measures.
- Identify obtuse triangles from side measures.
- Classify triangles in a number of different ways.
Converse of the Pythagorean Theorem
Could you use the Pythagorean Theorem to prove that a triangle contained a right angle if you did not have an accurate diagram?
You have learned about the Pythagorean Theorem and how it can be used. As you recall, it states that the sum of the squares of the legs of any right triangle will equal the square of the hypotenuse. If the lengths of the legs are labeled \begin{align*}a\end{align*} and \begin{align*}b\end{align*}, and the hypotenuse is \begin{align*}c\end{align*}, then we get the familiar equation:
\begin{align*}a^2+b^2=c^2\end{align*}
The converse of the Pythagorean Theorem is also true.
What is a converse?
A converse is an if-then statement where the hypothesis and the conclusion are switched.
For example, if we start with the if-then statement: If it is raining, then the street is wet.
The converse of our statement is: If the street is wet, then it is raining.
The part of the sentence after the “if” (called the hypothesis) switches places with the part of the sentence after “then” (called the conclusion.) You may remember these words from the previous chapters.
Statements and their converses are not always both true! In the case of the Pythagorean Theorem, BOTH the Theorem and its converse are true.
A converse is an if-then statement where the hypothesis and the __________________________ switch places.
Converse of the Pythagorean Theorem
Given a triangle with side lengths \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*} (where \begin{align*}c\end{align*} is the longest side), if the equation \begin{align*}a^2 + b^2 = c^2\end{align*} is true, then the triangle is a right triangle.
With this converse, you can use the Pythagorean Theorem to prove that a triangle is a right triangle, even if you do not know any of the triangle’s angle measurements.
This means that if you know the three side lengths of a triangle, you can substitute them into the equation \begin{align*}a^2 + b^2 = c^2.\end{align*}
- If you find a true statement (such as \begin{align*}100 = 100\end{align*}), then the Pythagorean Theorem works in that case; your triangle is a right triangle.
- If you find a false statement (such as \begin{align*}91 = 100\end{align*}), then, in that case, the Pythagorean Theorem does not work; your triangle is not a right triangle.
If \begin{align*}a^2 + b^2 = c^2\end{align*} produces a true statement, then the triangle is a ___________ triangle.
If \begin{align*}a^2 + b^2 = c^2\end{align*} produces a false statement, then the triangle is ___________ a right triangle.
Example 1
Does the triangle below contain a right angle?
This triangle does not have any right angle marks or measured angles, so you cannot assume you know whether the triangle is acute, right, or obtuse just by looking at it.
To see if the triangle might be right, try substituting the side lengths into the Pythagorean Theorem to see if they make the equation true. The hypotenuse is always the longest side, so 17 should be substituted for \begin{align*}c.\end{align*} The other two values can represent \begin{align*}a\end{align*} and \begin{align*}b\end{align*} and the order is not important.
\begin{align*}a^2 + b^2 &= c^2\\ 8^2 + 15^2 &= 17^2\\ 64 + 225 &= 289\\ 289 &= 289\end{align*}
Since both sides of the equation are equal \begin{align*}(289 = 289)\end{align*}, these values satisfy the Pythagorean Theorem. Therefore, the triangle described in the problem is a right triangle!
Reading Check:
1. Write the converse of this if-then statement:
If it is sunny outside, then the weather must be warm.
Converse:
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. BONUS! Write a true if-then statement that also has a true converse:
Statement:
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
Converse:
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
3. A triangle has side lengths 5, 7, and 9. Is this a right triangle? Show your work to defend your answer.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
Identifying Acute Triangles
If the sum of the squares of the two shorter sides of a triangle is greater than the square of the longest side, then the triangle is acute (all angles in the triangle are less than \begin{align*}90^\circ\end{align*}.)
If \begin{align*}a^2 + b^2 > c^2\end{align*} then the triangle is acute.
You can use this rule the same way you used the Pythagorean Theorem on the last page. Substitute the side lengths for \begin{align*}a, b,\end{align*} and \begin{align*}c\end{align*} making sure that the longest side is \begin{align*}c\end{align*}. After you have simplified both sides of the equation, compare your answers: which side is a larger number? If the \begin{align*}a^2 + b^2\end{align*} side is bigger than the \begin{align*}c^2\end{align*} side, then the triangle is acute.
- If \begin{align*}a^2 + b^2\end{align*} is larger than \begin{align*}c^2\end{align*}, then the triangle is ______________________.
Example 2
Is the triangle below acute or right?
The two shorter sides of the triangle are 8 and 13. The longest side of the triangle is 15.
Since the legs are the shorter sides, first find the sum of the squares of the two shorter legs by substituting the smaller numbers for \begin{align*}a\end{align*} and \begin{align*}b\end{align*}:
\begin{align*}8^2+13^2 & = c^2\\ 64 + 169 &= c^2\\ 233 &= c^2\end{align*}
The sum of the squares of the two shorter legs is 233. Compare this to the square of the longest side, 15.
\begin{align*}15^2=225\end{align*}
The square of the longest side is _____________.
Since \begin{align*}8^2 + 13^2 = 233\end{align*} and \begin{align*}233 \neq 225 = 15^2\end{align*}, this triangle is not a right triangle.
- \begin{align*}a^2 + b^2 \neq c^2\end{align*} so the triangle cannot be a ____________________ triangle.
Compare the two values to identify which is greater.
\begin{align*}233 >225\end{align*}
The sum of the squares of the shorter sides \begin{align*}(a^2 + b^2)\end{align*} is greater than the square of the longest side \begin{align*}(c^2)\end{align*}. Therefore, this is an acute triangle.
- Because \begin{align*}c^2\end{align*} is smaller than \begin{align*}a^2 + b^2\end{align*}, this is an ____________________ triangle.
Reading Check:
1. Fill in the blanks:
When the square of the __________________________ side is less than the sum of the squares of the _________________________ sides, the triangle is an acute triangle.
2. A triangle has side lengths 4, 7, and 8. Is this triangle acute or right? Show your work to defend your answer.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
Identifying Obtuse Triangles
You can prove a triangle is obtuse (meaning it has one angle larger than \begin{align*}90^\circ\end{align*}) by using a similar method.
Find the sum of the squares of the two shorter sides in a triangle. If this value is less than the square of the longest side, the triangle is obtuse.
If \begin{align*}a^2 + b^2 < c^2\end{align*}, then the triangle is obtuse.
- Obtuse triangles have one angle _____________________________ that is than \begin{align*}90^\circ\end{align*}.
- If \begin{align*}a^2 + b^2\end{align*} is smaller than \begin{align*}c^2\end{align*}, then the triangle is ______________________.
Example 3
Is the triangle below obtuse or right?
You can solve this problem in a manner almost identical to Example 2.
The two shorter sides of the triangle are 5 and 6. The longest side of the triangle is 10.
First find the sum of the squares of the two shorter legs by substituting the smaller numbers for \begin{align*}a\end{align*} and \begin{align*}b\end{align*}.
\begin{align*}a^2+ b^2 &= 5^2+ 6^2\\ &= 25+36\\ &= 61\end{align*}
The sum of the squares of the two shorter legs is 61. Compare this to the square of the longest side, 10.
\begin{align*}10^2=100\end{align*}
The square of the longest side is 100.
Since \begin{align*}5^2 + 6^2 = 61\end{align*} and \begin{align*}61 \neq 100 = 10^2\end{align*}, this triangle is not a right triangle.
Compare the two values to identify which is greater.
\begin{align*}61 & < 100\\ \text{(sum of shorter sides)}^2 & < \text{(longest side)}^2\end{align*}
Since the sum of the square of the shorter sides \begin{align*}(a^2 + b^2)\end{align*} is less than the square of the longest side \begin{align*}(c^2)\end{align*}, this is an obtuse triangle.
- Because \begin{align*}c^2\end{align*} is larger than \begin{align*}a^2 + b^2\end{align*}, this is an ____________________ triangle.
Reading Check:
1. Fill in the blanks:
When the square of the _______________________ side is greater than the sum of the squares of the ________________________ sides, the triangle is an obtuse triangle.
2. True or false: When the square of the longest side equals the sum of the squares of the shorter sides, the triangle is a right triangle.
\begin{align*}{\;}\end{align*}
3. A triangle has side lengths 5, 8, and 10. Is this triangle acute, obtuse, or right? Show your work to defend your answer.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
Triangle Classification
Now that you know the ideas in this lesson, you can classify any triangle as right, acute, or obtuse given the length of the three sides. Be sure to use the longest side for the hypotenuse.
Remember:
- If \begin{align*}a^2+b^2 = c^2\end{align*}, the figure is a right triangle.
- If \begin{align*}a^2+b^2>c^2\end{align*}, the figure is an acute triangle.
- If \begin{align*}a^2+b^2<c^2\end{align*}, the figure is an obtuse triangle.
Example 4
Classify the triangle below as right, acute, or obtuse.
The two shorter sides of the triangle are 9 and 11. The longest side of the triangle is 14. First find the sum of the squares of the two shorter legs by substituting the smaller numbers for \begin{align*}a\end{align*} and \begin{align*}b\end{align*}.
\begin{align*}a^2+ b^2 &= 9^2+ 11^2\\ &= 81+121\\ &=202\end{align*}
The sum of the squares of the two shorter legs is 202. Compare this to the square of the longest side, 14.
\begin{align*}14^2=196\end{align*}
The square of the longest side is 196. So the two values are not equal (\begin{align*}202 \neq 196\end{align*} or \begin{align*}a^2 + b^2 \neq c^2\end{align*}) and this triangle is not a right triangle.
Since you can eliminate the right triangle from your choices, now you can compare the two values, \begin{align*}a^2 + b^2\end{align*} and \begin{align*}c^2\end{align*} to identify which is greater:
\begin{align*}202 & > 196\\ \text{(sum of shorter sides)}^2 & > \text{(longest side)}^2\end{align*}
Since the sum of the square of the shorter sides is greater than the square of the longest side (in symbols \begin{align*}a^2 + b^2 > c^2\end{align*}), this is an acute triangle.
Example 5
Classify the triangle below as right, acute, or obtuse.
The two shorter sides of the triangle are _____________ and ______________.
The longest side of the triangle is ________________.
First, set up an equation to find the sum of the squares of the two shorter legs by substituting the smaller numbers for \begin{align*}a\end{align*} and \begin{align*}b\end{align*}.
\begin{align*}a^2+ b^2 &= 16^2+ 30^2\\ &= 256+900\\ &= 1156\end{align*}
The sum of the squares of the two legs is 1156.
Compare this to the square of the longest side, 34.
\begin{align*}c^2 = 34^2 =1156\end{align*}
The square of the longest side is also 1156.
Since the two values you found are equal \begin{align*}( a^2 + b^2 = c^2 )\end{align*}, this is a right triangle.
- In this example, the Pythagorean Theorem is ___________________.
- When \begin{align*}a^2 + b^2 = c^2\end{align*}, you have a ____________________ triangle!
Reading Check:
1. Fill in the blanks:
In an acute triangle, the sum of the squares of the shorter sides is _____________________ (greater than / less than / equal to) the square of the longest side.
In a right triangle, the sum of the squares of the shorter sides is _______________________ (greater than / less than / equal to) the square of the longest side.
In an obtuse triangle, the sum of the squares of the shorter sides is _____________________ (greater than / less than / equal to) the square of the longest side.
2. A triangle has side lengths 8, 9, and 13. Classify the triangle as right, acute, or obtuse. Show your work to defend your answer.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
Graphic Organizer for Lesson 6
Type of Triangle | Draw a picture | How can you use the Pythagorean Theorem to compare the sides? | Give an example of 3 side lengths for this triangle and show work to prove your classification |
---|---|---|---|
Right | \begin{align*}a^2 + b^2 = c^2\end{align*} | ||
Acute | |||
Obtuse |
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