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If the rectangular faces of a brick have their diagonals in the ratio 3 : 2√3 : √15, then the ratio of the length of the shortest edge of the brick to that of its longest edge is
- a)√3: 2
- b)1 :√3
- c)2 :√5
- d)√2: √3
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
If the rectangular faces of a brick have their diagonals in the ratio...
Assuming the dimensions of the brick are a, b and c and the diagonals are 3,2√3 and √15
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Hence, a2 + b2 = 32… (1)
b2 + c2 = (2√3)2…. (2)
c2 + a2 = (√15)2….. (3)
Adding the three equations, 2(a2 + b2 + c2) = 9+12+15=36
=>a2 + b2 + c2 = 18 …… (4)
Subtracting (1) from (4), we get c2 = 9 =>c=3
Subtracting (2) from (4), we get a2 = 6 =>a=√6
Subtracting (3) from (4), we get b2 = 3 =>b=√3
The ratio of the length of the shortest edge of the brick to that of its longest edge is =√3/3 =1:√3
Most Upvoted Answer
If the rectangular faces of a brick have their diagonals in the ratio...
Given Information:
The diagonals of the rectangular faces are in the ratio 3 : 2√3 : √15.
Let's assume:
Let the lengths of the diagonals be 3x, 2√3x, and √15x.
Diagonal of a Rectangle:
The diagonal of a rectangle can be found using the Pythagorean theorem.
Let the length, width, and height of the brick be a, b, and c, respectively.
Calculating the Diagonals:
- For the diagonal 3x:
- 3x² = a² + b²
- For the diagonal 2√3x:
- (2√3x)² = a² + c²
- For the diagonal √15x:
- (√15x)² = b² + c²
Ratio of Longest to Shortest Edge:
The longest edge of the brick is the diagonal of the rectangular face with the length 2√3x, and the shortest edge is the diagonal of the rectangular face with length 3x.
Solving for the Ratios:
- Longest edge: 2√3x
- Shortest edge: 3x
Ratio:
2√3x : 3x
2 : √3
Therefore, the ratio of the length of the shortest edge of the brick to that of its longest edge is 1 : √3, which corresponds to option 'b'.
The diagonals of the rectangular faces are in the ratio 3 : 2√3 : √15.
Let's assume:
Let the lengths of the diagonals be 3x, 2√3x, and √15x.
Diagonal of a Rectangle:
The diagonal of a rectangle can be found using the Pythagorean theorem.
Let the length, width, and height of the brick be a, b, and c, respectively.
Calculating the Diagonals:
- For the diagonal 3x:
- 3x² = a² + b²
- For the diagonal 2√3x:
- (2√3x)² = a² + c²
- For the diagonal √15x:
- (√15x)² = b² + c²
Ratio of Longest to Shortest Edge:
The longest edge of the brick is the diagonal of the rectangular face with the length 2√3x, and the shortest edge is the diagonal of the rectangular face with length 3x.
Solving for the Ratios:
- Longest edge: 2√3x
- Shortest edge: 3x
Ratio:
2√3x : 3x
2 : √3
Therefore, the ratio of the length of the shortest edge of the brick to that of its longest edge is 1 : √3, which corresponds to option 'b'.
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If the rectangular faces of a brick have their diagonals in the ratio 3 : 2√3 : √15, then the ratio of the length of the shortest edge of the brick to that of its longest edge isa) √3: 2b) 1 :√3c) 2 :√5d) √2: √3Correct answer is option 'B'. Can you explain this answer?
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