On Friday, we learned about The Pythagorean Theorem and Space Figures.
We mostly deal with the rectangular solid and regular square pyramid.
Rectangular Solid:
CDHG is one of the 6 faces.
is one of the 12 edges.
is one of the 4 diagonals of the rectangular solid.
If you were give the dimensions 8, 9, and 12 in order to find the diaganol
, it wouldn't matter which edges you assigned the lengths to. So you could let EG=9, GH=12, and AE=8. Then use the pythagorean theorem to find EH=15. Now from using AE=8 and EH=15, you can use the pythagorean theorem to solve for AH which equals 17.
Regular Square Pyramid:
The baseis a square.
The altitude is perpendicular to the base at its "center."
The slant height is perpendicular to the "side" of the base. To solve for the slant height, you would use pythagorean theorem with the height and half of the base.
Sorry this blog post was really short, but this section was short, so this pretty much covers it all.
Hello everyone!
Today we covered two sections, 9.6 & 9.7
Lets start with 9.6
9.6 was a pretty short, simple section about families of right triangles and Pythagorean Triples.
A Pythagorean Triple is any three whole numbers that satisfy the Pythagorean theorem,
For example, 3,4,and 5 are Pythagorean triples. Also, 8, 15, and 17
Any multiples of these are also in the same family of right triangles. For example 6, 8, and 10. These are all multiples of 3,4,and 5, therefore they are in the same family.
Next, is section 9.7
Section 9.7 has some shortcuts when dealing with "special" right triangles.
The first trick is for dealing with right triangles that have angle measures of 30°, 60°, and 90°.
Let a represent the side length of the short leg of the triangle. The hypotenuse is 2a. The long leg is a
For example, in the picture to the left, if a=4, the hypotenuse=8, and the long leg=4.
If you are slightly confused, I found that drawing an equilateral triangle and then drawing an altitude showed the relationships more clearly. The video below shows how to do this.
The video was giving me a little trouble so if it doesn't work here's the link
The next shortcut is for isosceles right triangles. These triangles have angle measures of 45°, 45°, and 90°. Let a represent the side length of the leg. Because it's isosceles the other leg is also a. The hypotenuse however, is a.
That wraps up sections 9.6 and 9.7. They were short and relatively easy sections. Personally I would prefer if our math homework looked like this
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 thatsimplifies 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
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!")
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.)
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. :)
(this guys name is trigon)
is one of the 12 edges.
For example, in the picture to the left, if a=4, the hypotenuse=8, and the long leg=4
or the pythagorean theorem
So area of green square + area of blue square = area of red square
