Geometry and Progressions! :)
In an isosceles right triangle, ABC,with the right angle at C, the length of a leg is 20 inches.Perpendiculars are dropped from C to the hypotenuse,from the foot of this perpendicular o leg CB,from the foot of this perpendicular to AB, then back to leg CB,etc. without end.Find the sum of the lengths of all the perpendiculars.
NOTE: write your answer in two four decimal places.
The answer is 48.2843.
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1 solution
the lengths of the perpendiculars are: 10 \sqrt{2}10,5 \sqrt{2}and so on. therefore the lengths of the perpendiculars are in geometric progression where r=1/\sqrt{2}, a_{1}=10\sqrt{2} ;
SUM= a_{1}/1-r; 20 * (\sqrt{2} + 1)= \boxed{48.28427} ;