Question

This question consists of two statements,namely,Assertion (A) and Reason (R).For selecting the correct answer,use the following code:

(a) Both Assertion (A) and Reason (R) are true and Reason (R) correct explanation of Assertion (A).

(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

(c) Assertion (A) is true and Reason (R) is false.

(d) Assertion (A) is false and Reason (R) is true.

$\begin{array}{cc}Assertion\left(A\right)& Reason\left(R\right)\\ \text{If ABCD is a rhombus whose one}& \text{Median of a triangle divides it}\\ \text{angle is}{60}^{\circ }\text{then the ratio of the}& \text{into two triangles of equal area.}\\ \text{lenghts of its diagonals is}\sqrt{3}:1.& \end{array}$

The correct answer is :(a) /(b)/(c)/(d).

Answer 1

(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
⇒∠ BDC = ∠ DBC = x°
Also, ∠ BCD = 60°
∴ x° + x° + 60° = 180°
⇒​ 2x° = 120°
⇒​ x° = 60°
∴ ∠ BCD = ∠ BDC = ∠ DBC = 60°
So, ​∆ BCD is an equilateral triangle.
i.e., BD = BC = a
∴ OB = $\frac{a}{2}$
Now, in ∆ OAB, we have:
$O{A}^2=A{B}^2-O{B}^2$

⇒$OA=\frac{\sqrt{3}a}{2}$

⇒ $AC=2\times \frac{\sqrt{3}a}{2}=\sqrt{3}a$
∴ $AC:BD=\sqrt{3}a:a=\sqrt{3}:1$
Thus, assertion (A)​ is also true, but reason (R) does not give (A).
Hence, the correct answer is (b).