Geometry/Chapter 5/Lesson 1

Introduction
In this lesson, we will review what a ratio is, in order to prepare you for the upcoming lessons.

What is a ratio?
A ratio is a comparison of two numbers or quantities. A ratio can be represented in four ways:
 * 1) As a fraction: $$\tfrac{a}{b}$$
 * 2) With a colon: a:b
 * 3) With the word "to": a to b
 * 4) With the word(s) "out of/over": a out of b or a over b

Example problem #1
If we had 4 oranges and 5 apples and wanted to answer the question: "What is the ratio of oranges to apples?"--how would we figure this out? We'd answer this question by figuring out the number of oranges and the number of apples. In this particular question, we have four oranges and five apples.

Now, we just plug in these numbers. But how? Simply look at the four ways a ratio can be represented. Let's pick number 3 to represent the ratio of oranges to apples.

Thus, the statement is: The ratio of oranges to apples is 4 to 5. Obviously, we can represent this value in the other ways that a ratio can be represented:
 * 1) $$\tfrac{4}{5}$$
 * 4:5
 * 1) 4 out of 5
 * 2) 4 over 5

#1 Rule
In ratios, we should always reduce or simplify ratios. If a ratio cannot be simplified, such 4:5 in the previous problem, then we leave it as is.

Example problem #2
In this next problem, we have 6 lemons and 8 pears. What is the ratio of lemons to pears? From seeing the total of lemons and pears, we can conclude our values to be 6 and 8. Are any of these answers correct?
 * 1) $$\tfrac{6}{8}$$
 * 6:8
 * 1) 6 to 8
 * 2) 6 out of 8/6 over 8

NO. NONE of these answers are correct. Although these answers are in the proper format, these values are not reduced! Always reduce ratios if you can!

So now, the true answer to this question can be answered in the following ways:
 * 1) $$\tfrac{3}{4}$$
 * 3:4
 * 1) 3 to 4
 * 2) 3 out of 4
 * 3) 3 over 4

Extended Ratios

 * Extended ratios - Extended ratios express two or more quantities with the "a:b:c:d" more (rule 2 above).

Example problem #3

 * 1) In a scalene triangle, the ratio of the angles of the triangle are $$8$$:$$3$$:$$9$$. Figure out the value of $$x$$ and the value of all the angles.


 * 1) All triangle angles, according to the Angle Sum Theorem, add up to 180. So, we have to equal all of our ratios to 180.

$$8x$$ + $$3x$$ + $$9x$$ = $$180$$
 * 1) We also add an $$x$$ to each number when we add it up to 180. So, our final equation would be:

$$20x$$ = $$180$$
 * 1) Combine all the $$x$$ terms together:

$$x$$ = $$9$$
 * 1) Divide $$180$$ by $$20$$, equalling $$9$$:


 * We have figured out the 1st part of our answer, which is $$9$$. Now, we need to figure out all the angles.

$$9(8)$$ = $$72$$ $$9(3)$$ = $$27$$ $$9(9)$$ = $$81$$
 * 1) Times $$9$$ by $$8$$, $$3$$, and $$9$$. Then you will get the sums of


 * We have figured out the 2nd part of our answer, which are $$72$$, $$27$$, and $$81$$.

OVERALL
The value of $$x$$ is $$9$$ and the value of all the angles (separately and in degrees) are $$72$$, $$27$$ and $$81$$. You can also fact-check this by checking to see if the angle values we've got add up to $$180$$.

$$72$$ + $$27$$ + $$81$$ $$99$$ + $$81$$  $$180$$