## Saturday, August 13, 2016

### Geometry Problem 1244: Circle, Radius, Perpendicular, Chord, Secant, Measurement. Mind Map

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to view more details of problem 1244.

1. https://goo.gl/photos/FGTGW9iSkgxRKfx16

Connect BF, BC and BD
Note that C is the midpoint of arc BG
Triangle BEC similar to FBC ( case AA)
So CB^2=CE.CF
Triangle DCB similar to BCA ( case AA)
So CB^2=CD.CA
And CE.CF=CD.CA or 4x 10= 3(x+3)
So x= 31/3

2. Let AB cut the circle at G.

In isoceles Tr. BCG,

CG^2 - CE^2 = BE.GE = CE.EF = 24
So CG^2 = 24 + 16 = 40

Now < CGB = CBG = < GDA

Hence < CGD = < DGB - < CGB = < DGB - < GDA = < GAD

Hence CG^2 = CD.CA

So 40 = 3(3+x) from whence x = 31/3

Sumith Peiris
Moratuwa
Sri Lanka

3. 2nd solution

< CFD = < CGD = < EAD as before

Hence AFED is cyclic and the result is easily calculated

Sumith Peiris
Moratuwa
Sri Lanka

4. Join points F.O.D then WE CAN eaisly solve for r.
Note that FO=OD=OC=r so Triangle FCD is a right triangle
By similar angle WE have traingle FCD is similar to ECA so
4/3+x=3/10 then X=31/3

1. First you will need to prove the collinearity of F.O.D