MathDB

A paper equilateral triangle

ABC

has side length

12

. The paper triangle is folded so that vertex

A

touches a point on side

\overline{BC}

a distance

9

from point

B

. The length of the line segment along which the triangle is folded can be written as

\frac{m\sqrt{p}}{n}

, where

m

n

, and

p

are positive integers,

m

and

n

are relatively prime, and

p

is not divisible by the square of any prime. Find

m+n+p

. [asy] import cse5; size(12cm); pen tpen = defaultpen + 1.337; real a = 39/5.0; real b = 39/7.0; pair B = MP("B", (0,0), dir(200)); pair A = MP("A", (9,0), dir(-80)); pair C = MP("C", (12,0), dir(-20)); pair K = (6,10.392); pair M = (a*B+(12-a)*K) / 12; pair N = (b*C+(12-b)*K) / 12; draw(B--M--N--C--cycle, tpen); draw(M--A--N--cycle); fill(M--A--N--cycle, mediumgrey); pair shift = (-20.13, 0); pair B1 = MP("B", B+shift, dir(200)); pair A1 = MP("A", K+shift, dir(90)); pair C1 = MP("C", C+shift, dir(-20)); draw(A1--B1--C1--cycle, tpen);[/asy]

9

Problems(2)