Three equal circles are drawn (see diagram).
A straight line connects the centers of the three circles.
Angle B is a right angle
What is the length of segment DE in terms of segment BC?
Hint: Let BC = 1, then solve for DE.
Solution to the Problem:
DE = 8/5 BC.
Dennis Beck sent in a different way to solve it using trigonometry:
angle DAF=arcsin(1/5)
angle ADF=180-arcsin(3/5)
angle FDE=arcsin(3/5)
angle DFE=arcsin(180-2 angle FDE)
DF=EF=1
Using the law of sines
Sin(angle FDE)/1=sin(angle DFE)/DE
DE=sin(angle DFE)/sin(angle FDE)
Sin(angle FDE)=3/5
Cos(angle FDE)=sqrt(1-(3/5)^2)=sqrt(1-9/25)=sqrt(16/25)=4/5
Sin(angle DFE)=sin(180-2 angle FDE)=sin(2 angle FDE)=2sin(angle FDE)cos(angle FDE)=2(3/5)(4/5)=24/25
DE=(24/25)/(3/5)=8/5=1.6
So DE=8/5(BC)
Correctly solved by:
| 1. James Alarie | Flint, Michigan |
| 2. Brian Morrison |
Thornton Academy, Saco, Maine |
| 3. Dennis Beck |
Clayton Valley Charter High School, Concord, California |
| 4. Vikash Kumar | New Delhi, India |