Geometry/Chapter 4/Lesson 6

Introduction

 * In this lesson, we will be reviewing the Pythagorean theorem. For more reading on this, see 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: The multiples of these numbers also work.
 * 1) $$3$$, $$4$$, $$5$$
 * 2) $$5$$, $$12$$, $$13$$
 * 3) $$8$$, $$15$$, $$17$$
 * 4) $$7$$, $$24$$, $$25$$
 * 5) $$9$$, $$40$$, $$41$$

For example, let's say a triangle has the following numerical inputs: ...and with this problem, we are asked:
 * $$a$$2 = 18
 * $$b$$2 = 24
 * $$c$$2 = 30

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

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



 * 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.
 * 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.
 * 1) Now that the radical has been solved, we simply plug in the numbers and solve:
 * 2) *$$2$$2 + $$12$$2 $$=$$ $$x$$2
 * 3) *$$4$$ + $$144$$ $$=$$ $$x$$2
 * 4) *$$148$$ $$=$$ $$x$$2
 * 5) Now that we have $$148 = x$$2, we need to square root these two factors.
 * 6) *√$$148$$ $$=$$ √$$x$$2
 * 7) After square rooting, we get $$x = 37$$√$$2$$

Answer: $$x = 37$$√$$2$$




 * $$21 - 11 = 10$$
 * 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




 * 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

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 $$41$$. As you can see, this Pythagorean triple is, indeed, a right angle.
 * Special Note
 * $$41$$2 = $$40$$2 + $$9$$2
 * $$1681$$ = $$1600$$ + $$81$$
 * $$1681$$ = $$1681$$