Posted by **harsh** on Tuesday, February 12, 2013 at 5:40am.

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?

- maths -
**Reiny**, Tuesday, February 12, 2013 at 8:14am
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

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