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:

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

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: Template:Collapse top

Answer: Yes, it is a Pythagorean triple of $3$, $4$ and $5$ Template:Collapse bottom ...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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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 $12$ will not be explained.
3. 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
4. Now that we have $148=x$², we need to square root these two factors.
   • √$148$ $=$ √$x$²
5. After square rooting, we get $x=37$√$2$

Answer: $x=37$√$2$

File:TrapezoidPythagoreanTheoremProblem.PNG

• $21-11=10$
• 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 Template:Collapse bottom

File:Trianglepythagoreantheoremproblem.PNG

• 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
• $400$ = $225$ + $169$
• $400$ = $394$
• $400$ > $394$

Answer: Obtuse triangle Template:Collapse bottom

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.

See also

• https://www.mathsisfun.com/pythagoras.html