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Three regular hexagons are drawn such that their diagonals cut each other at the same point and area A':A":A''' =1:2:3. Then the ratio of the length of the sides of the regular hexagons (from smallest to the largest) is?
Most Upvoted Answer
Three regular hexagons are drawn such that their diagonals cut each ot...
Solution:
Let the smallest hexagon have side length 'a', and the largest hexagon have side length 'c'. Then the side length of the middle hexagon is 'b'.
Area Ratio:
The area of a regular hexagon of side length 'x' is A = (3√3/2) x^2. Therefore, the area ratio is given by:
A':A":A''' = (3√3/2) a^2 : (3√3/2) b^2 : (3√3/2) c^2
= a^2 : b^2 : c^2
= 1 : 2 : 3
Therefore, b^2 = 2a^2 and c^2 = 3a^2.
Diagonal Ratio:
The diagonals of a regular hexagon of side length 'x' intersect at a point that is equidistant from all vertices. Therefore, the point of intersection of the diagonals of the three hexagons is the same. Let this point be O.
The diagonal of a regular hexagon of side length 'x' is 2x. Therefore, the ratio of the diagonals of the three hexagons is:
OA' : OA" : OA''' = 2a : 2b : 2c
= a : b : c
Therefore, b/a = A''/A' = 2/1 and c/a = A'''/A' = 3/1.
Final Ratio:
We have b^2 = 2a^2 and c^2 = 3a^2. Therefore, c^2/b^2 = 3/2. Substituting the value of c/a = 3/1, we get:
(3a)^2/b^2 = 3/2
b^2/a^2 = 4/3
Substituting the value of b/a = 2/1, we get:
(a/2)^2/a^2 = 4/3
a = √(3/4)
Therefore, the ratio of the side lengths of the three hexagons, from smallest to largest, is:
a : b : c
= √(3/4) : 2√(3/4) : 3√(3/4)
= √3 : √3(2) : √3(3)
= 1 : √3 : √3(3)
Hence, the required ratio of the length of the sides of the regular hexagons (from smallest to the largest) is 1 : √3 : √3(3).
Let the smallest hexagon have side length 'a', and the largest hexagon have side length 'c'. Then the side length of the middle hexagon is 'b'.
Area Ratio:
The area of a regular hexagon of side length 'x' is A = (3√3/2) x^2. Therefore, the area ratio is given by:
A':A":A''' = (3√3/2) a^2 : (3√3/2) b^2 : (3√3/2) c^2
= a^2 : b^2 : c^2
= 1 : 2 : 3
Therefore, b^2 = 2a^2 and c^2 = 3a^2.
Diagonal Ratio:
The diagonals of a regular hexagon of side length 'x' intersect at a point that is equidistant from all vertices. Therefore, the point of intersection of the diagonals of the three hexagons is the same. Let this point be O.
The diagonal of a regular hexagon of side length 'x' is 2x. Therefore, the ratio of the diagonals of the three hexagons is:
OA' : OA" : OA''' = 2a : 2b : 2c
= a : b : c
Therefore, b/a = A''/A' = 2/1 and c/a = A'''/A' = 3/1.
Final Ratio:
We have b^2 = 2a^2 and c^2 = 3a^2. Therefore, c^2/b^2 = 3/2. Substituting the value of c/a = 3/1, we get:
(3a)^2/b^2 = 3/2
b^2/a^2 = 4/3
Substituting the value of b/a = 2/1, we get:
(a/2)^2/a^2 = 4/3
a = √(3/4)
Therefore, the ratio of the side lengths of the three hexagons, from smallest to largest, is:
a : b : c
= √(3/4) : 2√(3/4) : 3√(3/4)
= √3 : √3(2) : √3(3)
= 1 : √3 : √3(3)
Hence, the required ratio of the length of the sides of the regular hexagons (from smallest to the largest) is 1 : √3 : √3(3).
Community Answer
Three regular hexagons are drawn such that their diagonals cut each ot...
Side is proportional area^2
thus 1:root 2 :root3
thus 1:root 2 :root3
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Three regular hexagons are drawn such that their diagonals cut each other at the same point and area A':A":A''' =1:2:3. Then the ratio of the length of the sides of the regular hexagons (from smallest to the largest) is?
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Three regular hexagons are drawn such that their diagonals cut each other at the same point and area A':A":A''' =1:2:3. Then the ratio of the length of the sides of the regular hexagons (from smallest to the largest) is? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Three regular hexagons are drawn such that their diagonals cut each other at the same point and area A':A":A''' =1:2:3. Then the ratio of the length of the sides of the regular hexagons (from smallest to the largest) is? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Three regular hexagons are drawn such that their diagonals cut each other at the same point and area A':A":A''' =1:2:3. Then the ratio of the length of the sides of the regular hexagons (from smallest to the largest) is?.
Three regular hexagons are drawn such that their diagonals cut each other at the same point and area A':A":A''' =1:2:3. Then the ratio of the length of the sides of the regular hexagons (from smallest to the largest) is? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Three regular hexagons are drawn such that their diagonals cut each other at the same point and area A':A":A''' =1:2:3. Then the ratio of the length of the sides of the regular hexagons (from smallest to the largest) is? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Three regular hexagons are drawn such that their diagonals cut each other at the same point and area A':A":A''' =1:2:3. Then the ratio of the length of the sides of the regular hexagons (from smallest to the largest) is?.
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