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?
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