A semicircle has AB as the endpoints of its diameter, and AB=400. Points C and D are on the circumference of the semicircle such that AD=BC=100. What is the length of DC?

Answer 1

let your centre be O.
Join OD and OC
Triangles AOD and BOD are clearly congruent (SSS)

let's look at triangle BOD, with sides 200, 200 , 100
and its central angle BOD = Ø
A "similar" triangle 2 , 2, 1 has the same angle Ø
by cosine law:
1^2 = 2^2 + 2^2 - 2(2)(2)cosØ
8cosØ = 7
cosØ = 7/8

It can be easily seen that CD || AB
so in triangle COD we have angle ODC = Ø
Let CD = 2x, draw an altitude from O to meet CD at P
clearly P is the midpoint, OPD is right-angled and
x/200 = cosØ
x = 200cosØ = 200(7/8) = 175
then 2x or CD = 350