MathDB

In the following diagram, a semicircle is folded along a chord

AN

and intersects its diameter

MN

B

. Given that

MB : BN = 2 : 3

and

MN = 10

. If

AN = x

, find

x^2

. [asy] size(120); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(pair P) { dot(P,linewidth(3)); return P; } real r = sqrt(80)/5; pair M=(-1,0), N=(1,0), A=intersectionpoints(arc((M+N)/2, 1, 0, 180),circle(N,r))[0], C=intersectionpoints(circle(A,1),circle(N,1))[0], B=intersectionpoints(circle(C,1),M--N)[0]; draw(arc((M+N)/2, 1, 0, 180)--cycle); draw(A--N); draw(arc(C,1,180,180+2*aSin(r/2))); label("

A

",D2(A),NW); label("

B

",D2(B),SW); label("

M

",D2(M),S); label("

N

",D2(N),SE); [/asy]

Problem Statement