The question consists of two statements, namely, Assertion (A) and Reason (R). Choose the correct answer.

Assertion (A): If ABCD is a rhombus whose one angle is $60 ∘$ then the ratio of the lengths of its diagonals is $3 : 1$ .

Reason (R): Median of a triangle divides it into two triangle of equal area.

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

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

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

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

answer is B.

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Detailed Solution

Given 2 statements.
Assertion (A): If ABCD is a rhombus whose one angle is

60 ∘

then the ratio of the lengths of its diagonals is

3 : 1

.
Reason (R): Median of a triangle divides it into two triangle of equal area.
Assertion: If ABCD is a rhombus whose one angle is 60°, - eSaral

As, all the sides are equal, using sides opposite to equal angle are equal,

∠ B D C = ∠ D B C = x

Now in triangle BDC, using angle sum property,
Question Image

Therefore, the angles are,

∠ B D C = 600 ∠ D B C = 600

Hence, BDC is an equilateral triangle.

B D = a B C = a

Now, using Pythagoras theorem in triangle OAB.

h y p o t e n o u s 2 = b a s e 2 + p e r p e n d i c u l a r 2 h y p o t e n o u s 2 − p e r p e n d i c u l a r 2 = b a s e 2

O A 2 = A B 2 − O B 2 O A 2 = a 2 − a 2 2 O A 2 = a 2 − a 2 4 O A 2 = 3 a 2 4

O A = 3 a 2

Now,

A C = 2 × O A A C = 2 × 3 a 2 A C = 3 a

The ratio is,

= A C : B D = 3 a : a = 3 : 1

The statement Assertion (A) is correct.
Now, for statement Reason (R)
The median divides the triangle having same base length and same height,
Hence, the area of both triangles so formed will be equal.
Statement Reason (R) is true.
But, Reason (R) is not correct explanation of Assertion (A).
Hence, option 2 is correct.

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