Question

In rhombus ABCD, diagonals AC and BD intersect each other at point O.
If cosine of angle CAB is 0.6 and OB = 8 cm, find the lengths of the side and the diagonals of the rhombus.

Answer 1

Consider the figure :

The diagonals of a rhombus bisect each other perpendicularly

cos ∠CAB = `(6)/(10) = (3)/(5)`

i.e.`"base"/"hypotenuse" = "OA"/"AB" = (3)/(5)`

Therefore if length of base = 3x, length of hypotenuse = 5x

Since
OB² + OA² = AB²  ...[ Using Pythagoras Theorem ]

(5x)² – (3x)² = OB²

OB² = 16x²

∴ OB = 4x

Now
OB = 8

4x = 8

x = 2

Therefore
AB = 5x

= 5 x 2

= 10 cm

And
OA = 3x

= 3 x 2

= 6 cm

Since the sides of a rhombus are equal so the length of the side of the rhombus 

The diagonals are
BD = 8 x 2

= 16 cm

AC = 6 x 2

= 12 cm