Question 457874
Kari has an analog clock, shown here, with a minute hand of length 12 in. When Kari’s clock strikes 2 o’clock, what is the area of the right triangle formed by drawing a segment from the end of the minute hand to the end of the hour hand?  Express your answer in simplest radical form.
<pre>
The angle between the hands at 2:00 is 2/12ths of 360°
or 1/6th of 360° or 60° 
{{{drawing(400,400,-16,16,-16,16,
line(0,0,0,12), circle(0,0,14), locate(-.5,13.5,12),locate(6sqrt(3)+.8,6+.8,2),
line(3sqrt(3),3)
)}}}{{{drawing(400,400,-16,16,-16,16,
line(0,0,0,12), circle(0,0,14), locate(-.5,13.5,12),locate(6sqrt(3)+.8,6+.8,2),
line(3sqrt(3),3), green(line(0,12,3sqrt(3),3), locate(.4,2.4,"60°"),
locate(-3,6,12in)))}}}

{{{drawing(400,400,-16,16,-16,16,
line(0,0,0,12), circle(0,0,14), locate(-.5,13.5,12),locate(6sqrt(3)+.8,6+.8,2),
line(3sqrt(3),3), green(line(0,12,3sqrt(3),3), locate(.4,2.4,"60°"),
locate(-3,6,12in),locate(2,1,6in)))}}}


Therefore the triangle is a 60°-30°-90° right triangle.  The hypotenuse
of such a right triangle is twice the shorter leg.  Therefore
the shorter leg is half of 12 in. or 6 in.

The green line is found by the Pythagorean theorem.

 c² = a² + b²
12² = 6² + b²
144 = 36 + b²
108 = b²
 b² = 108
       ___
  b = &#8730;108
       ____
  b = &#8730;36*3
        _
  b = 6&#8730;3


The formula for the area of a triangle is

   A = ½(base)(height)
                     _
base = 6, height = 6&#8730;3
              _
   A = ½(6)(6&#8730;3)
          _
   A = 18&#8730;3 square feet.

Edwin</pre>