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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
(2019)
- a)1 : √3
- b)√3 : 2
- c)√2 : √3
- d)2 : √5
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
If the rectangular faces of a brick have their diagonals in the ratio ...
Let the edges of the brick be a, b, and c such that
a < b < c
a2 + b2 = 32 = 9 ...(i)
a2 + c2 = ( 2√3 )2 = 12 ...(ii)
b2 + c2 = ( √15 )2 = 15 ...(iii)
Adding all three equations. We get
2(a2 + b2 + c2) = 9 + 12 + 15 = 36
a2 + b2 + c2 = 18 ...(iv)
From (i) and (iv), c2 = 18 – 9 ⇒ c = 3
From (iii) and (iv) a2 = 18 – 15 ⇒ a = 3
Therefore, required ratio =
a < b < c
a2 + b2 = 32 = 9 ...(i)
a2 + c2 = ( 2√3 )2 = 12 ...(ii)
b2 + c2 = ( √15 )2 = 15 ...(iii)
Adding all three equations. We get
2(a2 + b2 + c2) = 9 + 12 + 15 = 36
a2 + b2 + c2 = 18 ...(iv)
From (i) and (iv), c2 = 18 – 9 ⇒ c = 3
From (iii) and (iv) a2 = 18 – 15 ⇒ a = 3
Therefore, required ratio =
Most Upvoted Answer
If the rectangular faces of a brick have their diagonals in the ratio ...
Let the length, width, and height of the brick be l, w, and h, respectively.
According to the given condition, the diagonals of the rectangular faces have a ratio of 3:2.
The diagonal of a rectangular face can be found using the Pythagorean theorem. For the first face, the diagonal is √(l^2 + w^2).
For the second face, the diagonal is √(l^2 + h^2).
For the third face, the diagonal is √(w^2 + h^2).
Since the diagonals are in a ratio of 3:2, we can write the following equations:
√(l^2 + w^2) : √(l^2 + h^2) = 3 : 2 ...(1)
√(l^2 + h^2) : √(w^2 + h^2) = 3 : 2 ...(2)
Squaring both sides of equation (1), we get:
(l^2 + w^2) : (l^2 + h^2) = 9 : 4
Cross-multiplying, we have:
4(l^2 + w^2) = 9(l^2 + h^2)
4l^2 + 4w^2 = 9l^2 + 9h^2
5l^2 - 4h^2 = 4w^2 ...(3)
Similarly, squaring both sides of equation (2), we get:
(l^2 + h^2) : (w^2 + h^2) = 9 : 4
Cross-multiplying, we have:
4(l^2 + h^2) = 9(w^2 + h^2)
4l^2 + 4h^2 = 9w^2 + 9h^2
4l^2 - 9w^2 = 5h^2 ...(4)
From equations (3) and (4), we have the system of equations:
5l^2 - 4h^2 = 4w^2 ...(5)
4l^2 - 9w^2 = 5h^2 ...(6)
To find a specific solution, we need additional information or constraints.
According to the given condition, the diagonals of the rectangular faces have a ratio of 3:2.
The diagonal of a rectangular face can be found using the Pythagorean theorem. For the first face, the diagonal is √(l^2 + w^2).
For the second face, the diagonal is √(l^2 + h^2).
For the third face, the diagonal is √(w^2 + h^2).
Since the diagonals are in a ratio of 3:2, we can write the following equations:
√(l^2 + w^2) : √(l^2 + h^2) = 3 : 2 ...(1)
√(l^2 + h^2) : √(w^2 + h^2) = 3 : 2 ...(2)
Squaring both sides of equation (1), we get:
(l^2 + w^2) : (l^2 + h^2) = 9 : 4
Cross-multiplying, we have:
4(l^2 + w^2) = 9(l^2 + h^2)
4l^2 + 4w^2 = 9l^2 + 9h^2
5l^2 - 4h^2 = 4w^2 ...(3)
Similarly, squaring both sides of equation (2), we get:
(l^2 + h^2) : (w^2 + h^2) = 9 : 4
Cross-multiplying, we have:
4(l^2 + h^2) = 9(w^2 + h^2)
4l^2 + 4h^2 = 9w^2 + 9h^2
4l^2 - 9w^2 = 5h^2 ...(4)
From equations (3) and (4), we have the system of equations:
5l^2 - 4h^2 = 4w^2 ...(5)
4l^2 - 9w^2 = 5h^2 ...(6)
To find a specific solution, we need additional information or constraints.
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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(2019)a)1 : √3b)√3 : 2c)√2 : √3d)2 : √5Correct answer is option 'A'. Can you explain this answer?
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