LOGIN/REGISTER
Please Wait...
It is currently Sep 02, 2010, 11:32 am
Post new topic Reply to topic  Page 1 of 1
  [ 4 posts ]  Share: Facebook
Author Message
makar
Riemann Hypothesis
User avatar

Posts: 277
Location: India
India
Post Posted: Sep 13, 2009, 2:54 pm • # 1 


Let ABCD be a quadrilateral in which AB is parallel to CD and perpendicular to AD; AB = 3CD; and the area of the quadrilateral is 4. if a circle can be drawn touching all the four sides of the quadrilateral, find its radius.
 
 
thanhnam2902
Navier-Stokes Equations
User avatar

Posts: 1353
Location: Viet Nam
Viet Nam
Post Posted: Sep 14, 2009, 2:06 am • # 2 


This is my solution:

Let CD = x and radius of circle is R. Easy we get AB = 3x, BN = 2x, AD = 2R, AM = DQ = R

We have BC = BP + PC = BM + CQ = (AB + CD) - (AM + DQ) = 4x - 2R

Aply Pitago theorem for \triangle BCN we get BC^2 = NB^2 + CN^2

\Longleftrightarrow (4x - 2R)^2 = 4R^2 + 4x^2 \Longleftrightarrow (2x - R)^2 = R^2 + x^2 \Longleftrightarrow 3x^2 = 4Rx \Longleftrightarrow x = \frac {4}{3}R

But S_{ABCD} = 4 \Longleftrightarrow \frac {(AB + CD).AD}{2} = 4 \Longleftrightarrow \frac {4x.2R}{2} = 4 \Longleftrightarrow Rx ....

But x = \frac {4}{3}R \Longrightarrow R^2 = \frac {3}{4} \Longleftrightarrow R = \frac {\sqrt {3}}{2}


Attachments:
PVQ-CQT.PNG
PVQ-CQT.PNG [ 19.18 KiB | Viewed 35 times ]
 
 
makar
Riemann Hypothesis
User avatar

Posts: 277
Location: India
India
Post Posted: Sep 14, 2009, 11:35 am • # 3 


Join all the vertex of the quadrilateral with centre of the in-circle then area of the quadrilateral is
\frac{1}{2}R(AB+BC+CD+DA)=4
Also \frac {(AB + CD).AD}{2} = 4
Now solve for R (easy)
 
 
admire9898
Poincare Conjecture

Posts: 104
Location: LA,CA
Taiwan, Province of China    United States
Post Posted: Sep 15, 2009, 5:58 am • # 4 


thanhnam2902 wrote:
This is my solution:

Let CD = x and radius of circle is R. Easy we get AB = 3x, BN = 2x, AD = 2R, AM = DQ = R

We have BC = BP + PC = BM + CQ = (AB + CD) - (AM + DQ) = 4x - 2R

Hidden Text



good proof and easy to anybody to learn
 
 
Display posts from previous:  Sort by  
  Share on Facebook Previous topic | Next topic 

Community » Olympiad Section » Geometry » Geometry Unsolved Problems

All times are UTC - 8 hours [ DST ]
Share: Facebook

Moderators: yetti, MithsApprentice, N.T.TUAN, Peter, darij grinberg, orl, pohoatza, pbornsztein, High School Olympiad Moderators

Post new topic Reply to topic  Page 1 of 1
  [ 4 posts ] 

Login
Username:   Password:   Log me on automatically each visit  

You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot post attachments in this forum

Got popcorn?
AoPS has dozens of free videos, including the nationally-acclaimed MATHCOUNTS Minis.
Introduction to Geometry
A complete geometry course. Over 900 problems! Great for MATHCOUNTS and AMC preparation.