Assertion (A): Rhombus becomes square if its diagonals are equal. Reason (R): OB = OD = 1/2 x BD and OC = OA = 1/2 x AC ⇒ OB = OA = OC OB = OC ⇒ ∠a = ∠b = 45° Similarly, ∠c = ∠d = 45° ∠ABC = ∠b + ∠c = 45° + 45° = 90° 1. A is true, R is false. 2. A is false, R is true. 3. Both A and R are true. 4. Both A and R are false.

Mathematics

Assertion (A): Rhombus becomes square if its diagonals are equal.
Reason (R): OB = OD = $\dfrac{1}{2}\times \text{BD}$
and OC = OA = $\dfrac{1}{2}\times \text{AC}$
⇒ OB = OA = OC
OB = OC
⇒ ∠a = ∠b = 45°
Similarly, ∠c = ∠d = 45°
∠ABC = ∠b + ∠c = 45° + 45° = 90°
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.

Rectilinear Figures

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Both A and R are true.

Explanation

Properties of a Rhombus: The diagonals of a rhombus bisect each other at right angles.

The diagonals divide the rhombus into four right-angled triangles.

When the diagonals of a rhombus are equal, each of these right-angled triangles becomes an isosceles right triangle. This implies that all angles adjacent to the diagonals are 45°, making all four angles of the rhombus equal to 90°.

∴ Assertion (A) is true.

In the rhombus,

OB = OD = $\dfrac{1}{2}$ x BD

OC = OA = $\dfrac{1}{2}$ x AC

Since BD = AC,

⇒ OB = OD = OC = OA

Thus, OB = OC

In Δ OAB,

Let ∠ OAB = ∠ OBA = x°.

As the sum of all angles in a triangle is 180°,

∠ OAB + ∠ OBA + ∠ AOB = 180°

⇒ x° + x° + 90° = 180°

⇒ 2x° + 90° = 180°

⇒ 2x° = 180° - 90°

⇒ 2x° = 90°

⇒ x° = $\dfrac{90°}{2}$

⇒ x° = 45°

Hence, ∠ c = ∠ d = 45°

Similarly, ∠ a = ∠ b = 45°

Thus, ∠ ABC = ∠ a + ∠ b = 45° + 45° = 90°

∴ Reason (R) is true.

Hence, both Assertion (A) and Reason (R) are true.

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