MathDB

Triangle

ABC

is isosceles, with

AB=AC

and altitude

AM=11.

Suppose that there is a point

D

\overline{AM}

with

AD=10

and

\angle BDC=3\angle BAC.

Then the perimeter of

\triangle ABC

may be written in the form

a+\sqrt{b},

where

a

and

b

are integers. Find

a+b.

[asy] import graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-1.55,xmax=7.95,ymin=-4.41,ymax=5.3; draw((1,3)--(0,0)); draw((0,0)--(2,0)); draw((2,0)--(1,3)); draw((1,3)--(1,0)); draw((1,0.7)--(0,0)); draw((1,0.7)--(2,0)); label("

11

",(0.75,1.63),SE*lsf); dot((1,3),ds); label("

A

",(0.96,3.14),NE*lsf); dot((0,0),ds); label("

B

",(-0.15,-0.18),NE*lsf); dot((2,0),ds); label("

C

",(2.06,-0.18),NE*lsf); dot((1,0),ds); label("

M

",(0.97,-0.27),NE*lsf); dot((1,0.7),ds); label("

D

",(1.05,0.77),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

Problem Statement