A field in the shape of a right triangle (with the two shortest sides measuring 60 feet and 80 feet) has roads on all three sides that don't drain properly. As a result, muddy water puddles collect and when the cars pass through, they splash all the snow that is within 5 feet of any one of the three roads.
If all other snow not splashed by cars is kept clean, what percent is clean?
The inradius can be employed to great advantage here. The inradius of
the big triangle is (a+b-c)/2=20. The inradius of the similar
little triangle inside the big triangle is thus 20-5=15. The linear
proportionality factor is thus 15/20=3/4. The area ratio is thus
(3/4)²=9/16 or 56.25%.
The formula (a+b-c)/2 for the inradius r is easily deduced based on
calculating the area of the right triangle in two ways: as half of base
times height, or as the sum of the areas of the square and the two
"kites" at the vertices that are formed using the radii of the incircle
from its points of tangency with the triangle. Thus ab/2=
r(a-r)+r(b-r)+r² and solving the quadratic gives r=(a+b+c)/2 and
r=(a+b-c)/2. Only the smaller root can be the radius of the inscribed
circle, of course.
Edited on May 16, 2005, 12:43 am
Posted by Richard
on 2005-05-16 00:42:13