Geometry/Chapter 4/Lesson 6

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Introduction

• In this lesson, we will be reviewing the Pythagorean theorem. For more reading on this, see w:Pythagorean theorem.

Pythagorean theorem

The Pythagorean theorem is the world-wide famous geometric theorem that sets up the relationship between ${\displaystyle a}$2, ${\displaystyle b}$2 and ${\displaystyle c}$2 in a right triangle. ${\displaystyle c}$2 represents the hyptoenuse, or the longest side opposite of a right angle in a right-triangle. The formula is as described:

${\displaystyle a}$2 + ${\displaystyle b}$2 = ${\displaystyle c}$2

How do I use this theorem in a triangle problem?

First, it is important to note that the Pythagorean theorem has a few easy shortcuts to its geometric confusion. These are known as the Pythagorean triples. The triples are:

1. ${\displaystyle 3}$, ${\displaystyle 4}$, ${\displaystyle 5}$
2. ${\displaystyle 5}$, ${\displaystyle 12}$, ${\displaystyle 13}$
3. ${\displaystyle 8}$, ${\displaystyle 15}$, ${\displaystyle 17}$
4. ${\displaystyle 7}$, ${\displaystyle 24}$, ${\displaystyle 25}$
5. ${\displaystyle 9}$, ${\displaystyle 40}$, ${\displaystyle 41}$

The multiples of these numbers also work.

For example, let's say a triangle has the following numerical inputs:

• ${\displaystyle a}$2 = 18
• ${\displaystyle b}$2 = 24
• ${\displaystyle c}$2 = 30

...and with this problem, we are asked:

Do these lengths represent a right triangle?

Answer: Yes, it is a Pythagorean triple of ${\displaystyle 3}$, ${\displaystyle 4}$ and ${\displaystyle 5}$

...but, let's changed this equation. What about ${\displaystyle b}$2 is ${\displaystyle 25}$? Then this equation is not longer a Pythagorean triple, and therefore, we must plug in the numbers into the Pythagorean theorem equation:

• ${\displaystyle a}$2 + ${\displaystyle b}$2 = ${\displaystyle c}$2
• ${\displaystyle 18}$2 + ${\displaystyle 25}$2 = ${\displaystyle 30}$2
• ${\displaystyle 324}$ + ${\displaystyle 625}$ = ${\displaystyle 900}$
• ${\displaystyle 949}$ = ${\displaystyle 900}$

If ${\displaystyle 24}$ is replaced with ${\displaystyle 25}$, then we know that we have changed the triangle being dealt with from a right triangle to an obtuse triangle. See the section 2.3 for more info.

How do I use this theorem in a "find-x" problem?

How do we solve for "x" using the Pythagorean theorem?
1. We know that one of the properties of rectangles is that opposite sides are equal... so, knowing this, we can make the conclusion that the side opposite of 2√3 is also 2√3.
2. Now, we solve for the radical. Since this lesson is not about radicals, the explanation of how the answer is ${\displaystyle 12}$ will not be explained.
3. Now that the radical has been solved, we simply plug in the numbers and solve:
• ${\displaystyle 2}$2 + ${\displaystyle 12}$2 ${\displaystyle =}$ ${\displaystyle x}$2
• ${\displaystyle 4}$ + ${\displaystyle 144}$ ${\displaystyle =}$ ${\displaystyle x}$2
• ${\displaystyle 148}$ ${\displaystyle =}$ ${\displaystyle x}$2
4. Now that we have ${\displaystyle 148=x}$2, we need to square root these two factors.
• ${\displaystyle 148}$ ${\displaystyle =}$${\displaystyle x}$2
5. After square rooting, we get ${\displaystyle x=37}$${\displaystyle 2}$

Answer: ${\displaystyle x=37}$${\displaystyle 2}$

How do we solve for "x" using the Pythagorean theorem?
• ${\displaystyle 21-11=10}$
• Divide the product you have found, 10, by 2 and the bases of the 2 triangles next to the trapezoid measure to 5.
• Now that we know that our two triangles have a bottom of 5 and a hypotenuse of 13, we can make the conclusion that x is 12 from the Pythagorean triple: 5, 12, 13.

Answer: 12

How do we solve for "x" using the Pythagorean theorem?
• Divide 8 by 2. 4 is the value of the bottom leg of the two right triangles.
• Automatically we know x is 3 because of our Pythagorean triple: 3, 4, 5.

What is the converse of the Pythagorean Theorem?

Converse of the Pythagorean Theorem: If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

You can use the converse to determine if a triangle is acute, right or obtuse.

• Acute: ${\displaystyle c}$2 < ${\displaystyle a}$2 + ${\displaystyle b}$2
• Obtuse: ${\displaystyle c}$2 > ${\displaystyle a}$2 + ${\displaystyle b}$2
• Right: ${\displaystyle c}$2 = ${\displaystyle a}$2 + ${\displaystyle b}$2

If we have the numbers ${\displaystyle 13}$, ${\displaystyle 15}$ and ${\displaystyle 20}$, we automatically need to plug it into our Pythagorean Theorem equation.

• ${\displaystyle 20}$2 = ${\displaystyle 15}$2 + ${\displaystyle 13}$2
• ${\displaystyle 400}$ = ${\displaystyle 225}$ + ${\displaystyle 169}$
• ${\displaystyle 400}$ = ${\displaystyle 394}$
• ${\displaystyle 400}$ > ${\displaystyle 394}$
Now that we have worked out the problem: What triangle are we working with here?

Answer: Obtuse triangle

Special Note

We can use the converse of the Pythagorean Theorem to check if the Pythagorean triples are right angles. For example, let us use the triple ${\displaystyle 9}$, ${\displaystyle 40}$ and ${\displaystyle 41}$.

• ${\displaystyle 41}$2 = ${\displaystyle 40}$2 + ${\displaystyle 9}$2
• ${\displaystyle 1681}$ = ${\displaystyle 1600}$ + ${\displaystyle 81}$
• ${\displaystyle 1681}$ = ${\displaystyle 1681}$

As you can see, this Pythagorean triple is, indeed, a right angle.

See also

 Wikibooks has a book on the topic of Trigonometry/The_Pythagorean_Theorem.