Question 421332
<pre><font face = "consolas" color = "indigo" size = 4><b>


{{{drawing(320,200,-3,5,-1,4,

line(-2.16227766,0,4.16227766,0),
green(line(0,3,0,0)),
line(0,3,4.16227766,0),
locate(0.1,1.6,3),
line(-2.16227766,0,0,3),
locate(0,0,D),
locate(-2.3,0,A), locate(4.2,0,B), locate(0,3.4,C),
locate(-1.3,.5,x-2), locate(2,.5,x)

 )}}}

Right triangles ABC, ACD, and CBD are all similar.

Therefore since ACD and CBD are similar,

      x - 2     3
     ——————— = ———
        3       x

Cross multiply:

    (x - 2)x = 3*3
 
     x² - 2x = 9

 x² - 2x - 9 = 0

{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}

{{{x = (-(-2) +- sqrt( (-2)^2-4*(1)*(-9) ))/(2*(1)) }}}

{{{x = (2 +- sqrt(4+36 ))/(2*(1)) }}}

{{{x = (2 +- sqrt(40))/2 }}}

{{{x = (2 +- sqrt(4*10))/2 }}}

{{{x = (2 +- 2sqrt(10))/2 }}}

{{{x = (2(1 +- sqrt(10)))/2 }}}

{{{x = (cross(2)(1 +- sqrt(10)))/cross(2) }}}

{{{x = 1 +- sqrt(10) }}}

So DB has length {{{1+sqrt(10)}}} or about 4.162

CD is given to be 3, so since 3 is less than DB, which is 4.162.

Therefore CD is less than DB, choice 1)

Edwin</pre>