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, then the lengths of the side and the major & minor diagonals of the rhombus are respectively

• A. 12cm, 15cm, 22cm.
• B. 10cm, 16cm, 12cm.
• C. 30cm, 26cm, 32cm.
• D. 40cm, 36cm, 42cm.

Answer 1

The correct option is B 10cm, 16cm, 12cm.

We know that diagonal of rhombus are perpendicular to each other.

So angle AOB is 90 degree

So triangle AOB is right-angled triangle.

Let angle CAb =$a$

cos $a$=0.6

Given OB=8cm

tan $a$= $\frac{OB}{OA}$ -eq. 1

sin $a$ =$\frac{OB}{AB}$ -eq. 2

tan $a$= $\frac{sina}{cosa}$ -eq. 3

$\sin a=\sqrt{1-co{s}^{2}a}$ -eq. 4

By equation 4 we get

$\sin a=0.8$

By equation 3 we get

tan $a$=$\frac{4}{3}$

using equation 1 we get length of OA=6

By equation 2 we get length of AB =10

$\text{So Lenght of sides are 10 cm , length of major diagonal is 8x2=16 , length of minor diagonal =6x2=12}$

So correct answer is Option B