Chapter 9: The Distance Formula

Chapter 9: The Distance Formula

Hey guys, it's Robbie. Today in class we reviewed the distance formula.

One of the semi-new things we learned about the distance formula is that it comes directly from the Pythagorean Theorem.

http://www.purplemath.com/modules/distform.htm

Here is a link to a website that shows how this is. Basically, a right triangle is drawn with the legs falling on the lines of the coordinate plane. Then you can use the Pythagorean Theorem t o find the length of the hypotenuse. Using the distance formula on the

same triangle, you'll find that the Distance simplifies down to the the Pythagorean Theorem.

Ok, so the distance formula is:

So:

So, for example, you want to find the distance between (-4,5) and (-2,-1).

You do √(-2+4)² + (-1-5)². So that simplifies to √ 4+ 36 which simplifies to √40. This is pretty common with the Distance Formula. It results in a square root value a good number of times. To finish the example off √40 finally simplifies to 2 √10.

So, yeah, thats all we learned about today in class. Wasn't too difficult and mostly a review.

Robbie

P.S. The reason this is posted under Michael's account is because my account was not letting me post so I borrowed his.

P.S.S For those of you who play this, you lose the game

My epic blog post (filling in for Robbie)

Hello everyone its David, I'm filling in for Robbie. I hope thats okay.


Anyways today we learned about the pythagorean theorem and how there are many ways to prove it...

Here are the three ways we learned in class

The first way is...................

Start with a triangle

The area of one such triangle is of course 1/2ab (1/2Base x Height)
The area of four triangles identical to the original is 2ab (1/2ab x 4)
Now imagine it like this

The area of the large square is
The area of the small square is
so 
That simplifies to   or the pythagorean theorem

The next way to prove it is to start with another triangle

The area is still the same
Now picture this

The area of the large square is
And the area of the smaller square is 
 So you can set up the equation
Which again makes

*BONUS PROOF*

area of the green square is
area of the blue square is
area of the red square is So area of green square + area of blue square = area of red square




OR

 I lost the game :(
(Shane this is where you yell "I lost the game!")




David Mahoney

Section 9.1 and 9.2

9.1 Review of Radicals and Quadratic Equations

Quadratic Formula -

9.2 Introduction to Circles

An arc is made up of two points on a circle and all points of the circle need to connect those two points is a single path. On Circle D, arc BC is a minor arc, and is less than a semicircle. Arc BAC is a major arc, and is more than a semicircle.

Arcs can be measured in two ways, length and degrees. The measure of an arc is congruent to the measure of the angle formed by the radii that determine the points of the arc. (ex. 50 degrees)

To find the length of an arc, set up a proportion.

A sector is part of a disk. (ex. BDC) You also set up a proportion to find the area of a sector.

AB and AC are chords, and angle BAC is an inscribed angle on circle D. The measure of an inscribed angle is half of the arc it intercepts. (ex. angle BAC intercepts arc BC, measure of arc BC is 50, so angle BAC = 25 degrees.)

-Olivia Sheridan

Section 9.3

Today we learned about the Altitude-on-Hypotenuse theorems. In the first diagram of the picture above, you see a proof. This is proving the statement: if you draw a right triangle, and you draw altitude to hypotenuse, you create three similar right triangles.

You may now use this statement in your future proofs! YAY! :D
http://www.youtube.com/watch?v=KzBT8130TqU <-- this video shows you the different ways you can create these right triangles, changing the size's of the triangle. :)

http://www.youtube.com/watch?v=tuAjSiuG8j0
Watch it. Wait until after the advertisement. It's worth it...

OKAY. About the next diagram. This shows a right triangle with the altitude, the triangles next to it are the similar triangles formed by the altitude. There are many rules that ALWAYS apply to triangles under the "pink" rule (a.k.a the rule above in pink) These rules that are always true are in ORANGE.

You may notice there are also side lengths, found in the ratios, that are highlighted in blue. This was done NOT ONLY to *try* and make the diagram as cool as Mr. Wilhelm's, but for a learning stand point.


Highlighted in blue are the geometric means. This is relevant to Theorem 68- which, in short, says; The altitude to the hypotenuse is the mean proportional between segments of the hypotenuse.

The theorem also states; Either leg of the given right triangle is the mean proportional between the hypotenuse of the given right triangle and the segment of the hypotenuse adjacent to that leg.


(PS CLICK ON THE PICTURE, IT MAKES IT EASIER TO SEE!)

Hope you enjoy this, I tried. Failed, but ... "It is hard to fail, but it is worse never to have tried to succeed" Words of wisdom. :)

-Maggie Ridenour

QUIZ CANCELED!!

Sorry for the late notice, everybody.  I'm canceling the quiz.  We really need to have a test on Friday, so there's no room in the schedule to postpone it.  I hope you're not too disappointed.

Please spread the word.

-Mr. Wilhelm

p.s. -- The assignment from 8.5 is still due tomorrow.

yet another random blogviewer!

-Game (Tyler)

8.5: Three Theorems Involving Proportions

Hi everyone! Today in class, we learned three new theorems.

First is the side-splitter thereom:

            side-splitter: a line in a triangle that is parallel to one side and intersects the two other sides

                                 

            Proportion: 

Next is the theorem with parallel lines:

If three or more parallel lines are intersected by two transversals, the parallel lines divide the transversals proportionally.

Proportion:

 The last thereom is the angle bisector thereom:

If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides.

Other:

corollary-  a thereom that is a direct result of a previous one

That's all! Hope you liked the blog!

~Natalie