Page 1 of 1
Triangle Question
Posted:
Fri Apr 12, 2013 11:10 pm
by redhawk2015
Question #52 on 2013 Inv. D:
Triangle ABC exists such that m[unparseable or potentially dangerous latex formula] ACB = [unparseable or potentially dangerous latex formula] and point P is the midpoint of segment AB. If CP = [unparseable or potentially dangerous latex formula] cm, BC = [unparseable or potentially dangerous latex formula] cm, and m[unparseable or potentially dangerous latex formula] CPB = [unparseable or potentially dangerous latex formula], find [unparseable or potentially dangerous latex formula]
Answer: [unparseable or potentially dangerous latex formula] cm
I've tried a few approaches with Law of Sines/Cosines but can't seem to figure it out. Anyone know how?
Re: Triangle Question
Posted:
Fri Apr 12, 2013 11:28 pm
by qwerty123
The formula for the circumradius of a triangle is [unparseable or potentially dangerous latex formula].
This is a right triangle so [unparseable or potentially dangerous latex formula].
also [unparseable or potentially dangerous latex formula].
Let me use [unparseable or potentially dangerous latex formula] as the side b.
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula].
[unparseable or potentially dangerous latex formula]
Using the Law of Sines for [unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
Substitute in [unparseable or potentially dangerous latex formula] and R into the thing:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
Re: Triangle Question
Posted:
Sat Apr 13, 2013 11:51 am
by qwerty123
Or if you don't want to be overly-complicated like I did on my last post, just use the law of cosines once!
You know that [unparseable or potentially dangerous latex formula] because the circum-diameter of a right triangle is the hypotheneuse, so the radius is the midpoint segment of it.
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
Re: Triangle Question
Posted:
Sun Apr 14, 2013 7:54 pm
by 101dalmatians
LAW OF COSINES!!!!!!!!!!!!!!!!!!!!!! MY FAVORITE!!!!!!!!!!!!!!!!!!!!!! :P :D :O :W
Re: Triangle Question
Posted:
Mon Apr 15, 2013 2:54 am
by Fredfredburger
My method:
You know that [unparseable or potentially dangerous latex formula] is the geometric mean of [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula], and since [unparseable or potentially dangerous latex formula] we see that both must equal [unparseable or potentially dangerous latex formula]
That must mean [unparseable or potentially dangerous latex formula]
From there if you know your Pythagorean triples you can see that the the [unparseable or potentially dangerous latex formula]
Note that the since point P is a median, the area is cut in half, meaning the area of the smaller triangle is half of the entire area, or symbolically
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]