CAT 2019 Slot 1 — QA Question 3
Basics of Mensuration/PrismEasy
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?
Answer & solution
- A
√3 : 2
- B
2 : √5
1 : √3
- D
√2 : √3
Solution
Ratio of the three diagonals is 3 : 2√3 : √15
Let the legths of the three diagonals be 3k, (2√3)k and (√15)k.
And, the brick have length, breadth, height as x, y and z respectively.
∴ x2 + y2 = (3k)2 = 9k2 ...(1)
y2 + z2 = [(2√3)k]2 = 12k2 ...(2)
z2 + x2 = [(√15)k]2 = 15k2 ...(3)
Adding (1), (2) and (3), we get;
x2 + y2 + z2 = 18k2 ...(4)
Using (4) along with any of (1), (2) and (3), we get;
x = k√6 , y = k√3 and z = 3k,
Required ratio = (k√3)/3k = 1/√3.
Hence, option (c).