A parallelogram is divided into nine regions of equal area by drawing line segments parallel to one of its diagonals. What is the ratio of the length of the longest of the line segments to that of the shortest?
Answers
Given : A parallelogram is divided into nine regions of equal area by drawing line segments parallel to one of its diagonals
To Find : ratio of the length of the longest of the line segments to that of the shortest
Solution:
Let say ABCD is a parallelogram with Area = 9A
Each region area = 9A/9 = A
and AC is the diagonal , parallel to which line segments are drawn
Smallest line will form a similar triangle to ACD
Area of Δ ACD = 9A/2
Area of smallest triangle = A
k = shortest line segments
(Ratio of side of triangle )² = Ratio of area
=> (k/AC)² = A/(9A/2)
=> (k/AC)² = 2/9
=> k = AC √2/3
Area of 4th triangle = 4A
p = longest line segments
=> (p/AC)² = 4A/(9A/2)
=> (p/AC)² = 8/9
=> p = AC 2√2/3
p / k = AC 2√2/3 / AC √2/3
=p/k = 2
ratio of the length of the longest of the line segments to that of the shortest = 2
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