Assertion: If $A B C D$ is a rhombus whose one angle is $60^{\circ}$, then the ratio of the lengths of its diagonals is $\sqrt{3}: 1$.
Reason: Median of triangle divides it into two triangles of equal area.
(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.
(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
Reason (R) is clearly true.
The explanation of assertion (A) is as follows:
ABCD is a rhombus. So, all of its sides are equal.
Now, BC = DC
$\Rightarrow \angle B D C=\angle D B C=x^{\circ}$
Also, $\angle B C D=60^{\circ}$
$\therefore x^{\circ}+x^{\circ}+60^{\circ}=180^{\circ}$
$\Rightarrow 2 x^{\circ}=120^{\circ}$
$\Rightarrow x^{\circ}=60^{\circ}$
$\therefore \angle B C D=\angle B D C=\angle D B C=60^{\circ}$
So, ∆BCD is an equilateral triangle.
i.e., BD = BC = a
$\therefore O B=\frac{a}{2}$
Now, in $\triangle \mathrm{OAB}$, we have:
$O A^{2}=A B^{2}-O B^{2}=a^{2}-\left(\frac{a}{2}\right)^{2}=\frac{3 a^{2}}{4}$
$\Rightarrow O A=\frac{\sqrt{3} a}{2}$
$\therefore A C: B D=\sqrt{3} a: a=\sqrt{3}: 1$
Thus, assertion (A) is also true, but reason (R) does not give (A).
Hence, the correct answer is (b).
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