Sections 8.1 and 8.2

Hey guys!  Sorry this didn't go up yesterday, hope you didn't have to much trouble on the homework...

Also, heres a funny video about ratios if you watch it all the way through. Ratios: Bad Date

After your done reading,  Click here for a new video from the makers of one dozen monkeys!

8.1: Ratios and Proportions

First of all, let's start with some basic definitions that boil these concepts down.

Ratio: Comparison of two numbers (Using fraction or colon).

Example: 5:6 or 5/6

Proportion: Setting two ratios equal to each other.

Example:
  

<----That form will be the basis for all the properties discussed below.

Oh, and also here are two necessary definitions.

		Means: Second and third terms (b and c)
		Extremes: First and fourth terms (a and d)

  

Now that we have the basics out of the way, let's talk about some of the properties of proportions.

               
 Theorem 59: In a proportion, the product of the means is equal to the products of the extremes.

   ad=bc

This theorem uses basic cross-multiplying to simplify proportions.

Other theorems summed up to the following simple equations:


  






And also, an unexpected, yet true statement,





One last portion of this chapter included a new concept: arithmetic and geometric means.

The arithmetic mean of two numbers is found using the following equation:




While on the other hand, the geometric mean of two numbers is found using this equation:





That concludes Section 8.1.


8.2: Introduction to Similarity

In simple terms, similar figures are figures that proportionally have the same shape, but are not necessarily the same size.

In order for two polygons to be considered "similar", the must have one of two basic qualities:

-All corresponding angles congruent
-All corresponding side lengths are proportional













Some other terms that were also mentioned in the book were:

Dilation: An enlarged figure that is similar to the original figure.

Reduction: A figure reduced in size that is similar to the original figure.

This problem shows an example of a common problem which ties in similar figures, ratios, and proportions.

Another theorem that was also briefly mentioned in the book is as follows:

Theorem 61: The ratio of the perimeters of two similar polygons equals the ratio of any pair of corresponding sides.

This means that if, for example, the perimeter of an original square is 20 units, and the perimeter of the dilation of the square is 40 units, than the ratio of the perimeters as well as the ratios of corresponding sides should be equal.


That is about it for today's information! Thanks, and hoped it helped!



Juliaaaaaaa