CAT Exam > CAT Questions > Three regular hexagons are drawn such that th...
Start Learning for Free
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
Attention CAT Students!
To make sure you are not studying endlessly, EduRev has designed CAT study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in CAT.
|
Explore Courses for CAT exam
|
|
Top Courses for CATView all
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?
Question Description
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?.
Solutions 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? in English & in Hindi are available as part of our courses for CAT.
Download more important topics, notes, lectures and mock test series for CAT Exam by signing up for free.
Here you can find the meaning of 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? defined & explained in the simplest way possible. Besides giving the explanation of
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?, a detailed solution 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? has been provided alongside types of 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? theory, EduRev gives you an
ample number of questions to practice 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? tests, examples and also practice CAT tests.
|
|
Explore Courses for CAT exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup