What else can be used to show them and clear the concepts?

Parallel: Two railways, cables.

Perpendicular: The corner of the classrooms, a cross.

Wow! Thank. That is soooo funny. Talk about a revelation. Thank you so much. I wonder what a venn diagram or euler might look like for this.

Philosophers would be the universe set. Absent-minded if the set inside that universe set and Zitta N is outside of it.

]]>Thanks,

ET ag

bob bundy wrote:Imagine rotating triangle TAC through 90 degrees, anticlockwise, around point A.

T rotates to B and C rotates to V, so one triangle becomes the other.

So TC must also rotate 90 to become BV, hence they are perpendicular.

Bob

How do we know that you have to rotate it 90 degrees?

Well, if we wanted to rotate point

anti-clockwise about point , it would end up at point This is because is thus must "travel" that angle to reach point .(Sorry if the LaTeX doesn't seem very neat, I tried to make it "proper" using the correct commands on this database. Quite ironic how LaTeX is supposed to make math look neater when in fact it makes it look quite messy on this website, if you ask me).

(Funny how I have edited this message like 20 times and I have only sat at my desktop for about eight minutes).

From,

ET ag

I use capital letters for the point location and small letters for position vector. that always gives a disciplined approach rather than using free vectors.the origin acts as anchor so that you won't go astray.]]>

Welcome to the forum.

The forum LaTex doesn't support your draw commands so you won't get an image that way. Most members use Geogebra to construct their diagrams; save them to an on-line site; and then link from there using bcCode. A recent post has addressed this.

Bob

]]>Since you use AoPS, you should be familiar with the AoPS Honor Code. Specifically, you should know that you should try to look up answers but rather try them. Remember, the teachers of your course are happy to help you on the Page Feed.

Also, the answer 0 is correct given Bob's reason.

]]>f(x) o f(x)^-1 = f(x)^-1 o f(x) = identity operation (same as if no operation was performed) = x]]>

hi Maroon900

Welcome to the forum.

At first I thought this was a use of the triangle inequality and, after a bit of manipulating, I got AD < AB + AC. Nearly there I thought. Then I tried a fresh approach:

Rotate the triangle about point D to create a new triangle A'BC below the first. The resulting shape ABA'C is a parallelogram (can you prove this?) and the diagonal is 2AD. You'll get the required result from this.

Bob

Brilliant. Creating parallelogram was the trick.

Maroon900, Hint, SSS and 2AD < AC+ A'C

]]>"Jayne has 3 quarters, 2 dimes, a nickel, and 2 pennies in her pocket. How many different amounts can she make using some or all of these coins?" Please solve this and show me how you got the answer step-by-step.

Use generating functions:

But we need to use some of the coins so 0 is out. There are 71 ways to give change using these coins but only 62 of them are different amounts so that is the answer.

]]>JaneFairfax wrote:

Twenty-three. Thats one name for the number 23.it is numbers not STUPID LETTERS

13kayduz wrote:

Shurrik wrote:二十三 = 23 (in Chinese)

Thank you oh so very much:D

Those are letters too. --'

]]>Sorry, but I can't answer your ranking question. Maybe someone else here can.

I googled around a bit and came across quite a range of coverage about this topic on the net. Here are some search suggestions:

"Sports teams' ranking"

"Sports rating system methods"

"Sports power ranking"

Hopefully a ready-made system exists for your model!

]]>Yes, you are correct. I got lazy as it's 1/2 on both choices so it doesn't effect the final answer.

Bob

]]>Of course the maximum is a lot more difficult...]]>