Geometry/Chapter 4/Lesson 6

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 $a$ ², $b$ ² and $c$ ² in a right triangle. $c$ ² represents the hyptoenuse, or the longest side opposite of a right angle in a right-triangle. The formula is as described:

$a$ ² + $b$ ² = $c$ 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:

$3$ , $4$ , $5$
$5$ , $12$ , $13$
$8$ , $15$ , $17$
$7$ , $24$ , $25$
$9$ , $40$ , $41$

The multiples of these numbers also work.

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

$a$ ² = 18
$b$ ² = 24
$c$ ² = 30

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

Do these lengths represent a right triangle?
Answer: Yes, it is a Pythagorean triple of $3$ , $4$ and $5$

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

$a$ ² + $b$ ² = $c$ ²
$18$ ² + $25$ ² = $30$ ²
$324$ + $625$ = $900$
$949$ = $900$

If $24$ is replaced with $25$ , 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?

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How do we solve for "x" using the Pythagorean theorem?

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.
Now, we solve for the radical. Since this lesson is not about radicals, the explanation of how the answer is $12$ will not be explained.
Now that the radical has been solved, we simply plug in the numbers and solve:
- $2$ ² + $12$ ² $=$ $x$ ²
- $4$ + $144$ $=$ $x$ ²
- $148$ $=$ $x$ ²
Now that we have $148=x$ ², we need to square root these two factors.
- √ $148$ $=$ √ $x$ ²
After square rooting, we get $x=37$ √ $2$

Answer: $x=37$ √ $2$

How do we solve for "x" using the Pythagorean theorem?

$21-11=10$
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: $c$ ² < $a$ ² + $b$ ²
Obtuse: $c$ ² > $a$ ² + $b$ ²
Right: $c$ ² = $a$ ² + $b$ ²

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

$20$ ² = $15$ ² + $13$ ²
$400$ = $225$ + $169$
$400$ = $394$
$400$ > $394$

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 $9$ , $40$ and $41$ .

$41$ ² = $40$ ² + $9$ ²
$1681$ = $1600$ + $81$
$1681$ = $1681$

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