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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.
Solution
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
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