Problems v.1

[darksaber21]

Okay, going through different tests, I found three questions I need help with (it's always the stated on the fourth page that trips me up -_-):

Edit - Alright, got them all taken care off now. So thanks to all you guys.

I was going to ask for help on another question, but I figured it out at the last minute an hour or so ago, so I'm fine there.

Thanks! I really appreciate it guys!

[nsguy1350]

Fourth page stated and geometry problems often turn into study guide problems for future years. And some of those problems are TOUGH, so don't worry about them too much.

04B-70: That's not enough information.

11D-38: Last year, we came up with a 2012 Stated Problem Study Guide thread. We solved, checked, and posted our solutions in formula format so that we can solve the problems quickly.

I believe the formula for this problem was [unparseable or potentially dangerous latex formula], where [unparseable or potentially dangerous latex formula] is the longest side and [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are the shorter sides. Plugging in [unparseable or potentially dangerous latex formula], we get [unparseable or potentially dangerous latex formula].

12B-37 seems like a lot of work. I am not sure exactly how to incorporate length into the solution, so I'll just leave it alone for now.

[FabensMath]

Hexagon, hint: to get a trick work yourself backwards from answer to trick, keeping in mind the proportions to area and lengths.
12B-37 was posted before, the solution was by Shawn in a topic called Conglomeration of Random Questions i believe so check the previous topics, and nsguy is right, if "going through different tests" means going through study guide, you might want to wait a while so we start a thread with ordered solutions, but it's kinda early for that in my opinion but it's just me. :)

[darksaber21]

Okay, thanks guys. And yeah, I was looking through the study guide, and through other tests. 'Those were ones I was was stuck on for a while. But anyway, I'm still going to try to solve these so I can move on to other questions.

There are still questions I don't understand, but I'll only ask if I can't figure them out in a couple of days. But yeah, thanks again for your input.

Edit - I found out where I was going wrong for the potato question. I apparently wasn't think straight. So onward to solving the others!

[88bobcat]

[88bobcat]

[unparseable or potentially dangerous latex formula]


In this derivation:

(1) [unparseable or potentially dangerous latex formula], [unparseable or potentially dangerous latex formula], and [unparseable or potentially dangerous latex formula] denote the original triangular spacing; and

(2) [unparseable or potentially dangerous latex formula] denotes the distance to move to create a right triangle; and

(3) [unparseable or potentially dangerous latex formula] is a percentage that determines the location of cevian [unparseable or potentially dangerous latex formula].


Applying Stewart's Theorem twice:

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]


However, [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are related:

[unparseable or potentially dangerous latex formula]


Solving the first equation for [unparseable or potentially dangerous latex formula], and substituting into the second equation yields:

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]


Expanding this second equation:

[unparseable or potentially dangerous latex formula]


[unparseable or potentially dangerous latex formula] is also determined by applying Stewart's Theorem:

[unparseable or potentially dangerous latex formula]


Solving for [unparseable or potentially dangerous latex formula]:

[unparseable or potentially dangerous latex formula]


Substituting for [unparseable or potentially dangerous latex formula] and partially for [unparseable or potentially dangerous latex formula] and rearranging:

[unparseable or potentially dangerous latex formula]


This can be expressed as

[unparseable or potentially dangerous latex formula]


Solving this quadratic for [unparseable or potentially dangerous latex formula] yields:

[unparseable or potentially dangerous latex formula]


To minimize [unparseable or potentially dangerous latex formula], a criterion is [unparseable or potentially dangerous latex formula]:


As it turns out, [unparseable or potentially dangerous latex formula] is minimized when [unparseable or potentially dangerous latex formula], meaning that [unparseable or potentially dangerous latex formula] is a median.


[unparseable or potentially dangerous latex formula]

[darksaber21]

Makes perfect sense now. Thanks. I knew I had to use calculus at some point to find the minimum distance, but I never would've thought to use Stewart's theorem twice.
And I haven't seen it on any test this year as far as I know. I don't know anything else past that.

[88bobcat]