SEN POTW #1

Weekly, I will post a mathematical question to be solved by us amazing SEN'ers! The problems may vary wildly, and most will be rather difficult to solve. Depending on difficulty, I will award more or less points than previous POTW's! A scoreboard will be kept in a collapsible box at the bottom of each POTW! Hints may or may not be given depending on the question, and will never affect the points to be awarded! The points will be awarded to the first person to answer the POTW correctly, guessing is allowed ~at your own risk! Work must be shown upon giving your answer, just so that other participants can see your thoughts and possibly learn from them. If sufficient work isn't shown then you may be awarded less points than stated for that particular POTW!

Without further ado, here is POTW #1

Points to be awarded upon a correct answer: 5

"An isosceles triangle has 2 circles laid within its three sides, and the circles are each tangent to the edges of the triangle, as well as each other. One has a radius of 1, and the other a radius of 2. What is the area of the entire triangle? Note: Picture may not be drawn to scale."






Post has been edited 4 time(s), last time on Oct 1 2009, 3:45 am by CecilSunkure.

Rawr, is it about 15.7?

It sadly, != 15.7..

This problem took me 1 full page of tinkering until I finally realized how to solve it, then solved it in about 1/4 of a page, without a calculator.

16.52

Hint before I go to bed:

(1 + x)/1=(2+x)/2

Good Luck!

Been a while since Geometry. Guessing you use ?

Refer to my attached labels.

ACK ~ AFM. (share a common angle A and each has another right angle (C, F), therefore similar)
Note MF/KC = 2
=> AM/AK = 2
Since KM = 3, AK = 3.
=> AH = 8 and AC = sqrt(8) (pythag. thm)
Also ACK ~ AHI. (same reason)
=> AC/CK = AH/HI
=> sqrt(8)/1 = 8/HI
=> HI = sqrt(8).

The area is AH*HI = 8*sqrt(8) = 16*sqrt(2)

Attachments:

Oops. When finding the angle between the height and one of the sides I used 2/8 instead of 2/6

Not following you on this step. How do you know that?

Nice, basically, to solve this you set up two simple ratios to solve for x. He did what I did in my hint to solve for X, which gave you a height of two from highest point on the top circle, to the highest point on the triangle. That meant that 4, 2, 2, resulted in a hieght of 8. From here you can use the Pythagorean Theorem to find side lengths of smaller triangles within the largest one (Like AC) - allowing you to see that triangle ACK ~ AHI (similar). You set up another ratio and cross multiply, solving for the base.

Then, it's 1/2 base * height.

Congratz to azala!

[Edit]I'll be sure to add labels to the points on shapes for the next POTW.

Post has been edited 1 time(s), last time on Sep 30 2009, 1:33 pm by CecilSunkure.

If we know:
ACK ~ AFM.
MF/KC = 2

In a similar triangle side ratios hold.
So AM/AK = AC/AF = MF/KC

Cecil your "hint", (1 + x)/1=(2+x)/2, was rather worthless. It implies x = 0, and you didn't even define x. I just looked at the triangle, labeled everything, and said "slew of similar triangles" and then attempted to find the most ruthlessly efficient set of ratios to solve it. These types of problems almost never involve finding angle measures.
I suppose it's a bit unfair, since I'm a coach of the Minnesota ARMLteam. I'll just PM solutions next time

Post has been edited 6 time(s), last time on Sep 30 2009, 8:37 pm by azala.

PotW should include biology/chemistry problems. I'm good at those.
Also, can you put a link to the post(s) with the correct answer and explanation in the OP?

So are these every Tuesday, or whenever Cecil thinks of some random problem from a high-schooler's math homework?

Sure, I didn't define x, and it does imply that x = 0. That was due to a typo.. Sorry about that.

(1 + x)/1=(4+x)/2 is what I meant to write, leaving x = 2.

I really didn't mean to make it look like I was taking credit away from you solving the problem on your own or anything, and the hint didn't define x by design; I didn't want to give anything away easily.. Yet.

And yes, I can quote the person who solved the problem correctly in the OP as well as explaining a few things.

The problems I get aren't from my homework, they are from a collection of problems my Calculus teacher has. He gives them to us to solve for extra credit.

I'll consider adding Bio and Chem questions, though I'm not sure if I'll follow through with it.

I'll be posting these every Monday//Tuesday ish time.

physics questions too pls.

No I didn't think you were taking credit. It just seemed like a very strange hint.


Then that's not really fair to the rest of your class, is it?

Hmm? He allows any student in any math class of his to try to solve the problem, and collaborate with anyone from any class for help. I or a friend will usually solve it, then help others with it. If we all are stumped, then our teacher gives us some hints to help us out. The POTW at my school is more of to the give students who want to put the effort into getting the extra credit, extra credit. Most kids think it isn't worth spending time on or too nerdy, w/e, and as such most never bother with the POTW.

I'll think about it.

I'm thinking I'll have the POTW always be a math question, with an extra question or two of other subjects for 1 point each, added onto the post.

Does that sound good?

What he's saying is.. by asking the internet, you're getting an "unfair" advantage over your peers despite the fact that they have internet as well so really, who the fuck cares?
Also, your one question is boring and I suspect all future questions will be at the same boringness level. I understand that I am not forced to do them so this is my fault for reading it. That doesn't make it any less boring though.

Oh, I won't post questions unless I know the answer.

I find these rather fun. I enjoy math as it is a really useful tool for all sorts of things extremely relevant to life in general. Also, the competitive feel to the POTW makes it all the more enjoyable

Yeah, because we totally trust you won't "post questions" unless you "know the answer".