Question

The sides of a rhombus ABCD measure 2 cm each, and the difference between two of its angles is \(90^{\circ}\). Then the area of the rhombus is:

• A. \(\sqrt{2}\) sq cm
• B. \(2\sqrt{2}\) sq cm
• C. \(3\sqrt{2}\) sq cm
• D. \(4\sqrt{2}\) sq cm

Hint:

A rhombus has only two distinct angles that add up to 180 degrees; use their given difference to find each one, then apply Area = side squared times sine of the angle.

Answer 1

The Correct Option is B

Step 1: Find the two distinct angles of the rhombus.
Adjacent angles of a rhombus add up to $180^{\circ}$, so if one angle is $\theta$, the next one is $180^{\circ}-\theta$. Their difference is given as $90^{\circ}$:
\[ (180^{\circ}-\theta)-\theta = 90^{\circ} \]
\[ \theta = 45^{\circ} \]
So the rhombus has angles of $45^{\circ}$ and $135^{\circ}$.

Step 2: Recall how the diagonals relate to the side and the angle.
The diagonals of a rhombus bisect each other at right angles and also bisect the vertex angles. At the vertex with angle $45^{\circ}$, the diagonal through it splits that angle into two parts of $22.5^{\circ}$ each. In the right triangle formed by half of each diagonal and a side of length $2$ cm:
\[ \frac{p}{2} = 2\sin(22.5^{\circ}), \qquad \frac{q}{2} = 2\cos(22.5^{\circ}) \]
where $p$ and $q$ are the lengths of the two diagonals.

Step 3: Write the diagonals explicitly.
\[ p = 4\sin(22.5^{\circ}), \qquad q = 4\cos(22.5^{\circ}) \]

Step 4: Use the diagonal formula for the area of a rhombus.
\[ \text{Area} = \frac{1}{2}pq = \frac{1}{2}\times 4\sin(22.5^{\circ}) \times 4\cos(22.5^{\circ}) = 8\sin(22.5^{\circ})\cos(22.5^{\circ}) \]

Step 5: Apply the double angle identity to simplify.
Using $2\sin\alpha\cos\alpha = \sin(2\alpha)$ with $\alpha = 22.5^{\circ}$:
\[ 8\sin(22.5^{\circ})\cos(22.5^{\circ}) = 4 \times \big(2\sin(22.5^{\circ})\cos(22.5^{\circ})\big) = 4\sin(45^{\circ}) \]
\[ = 4 \times \frac{\sqrt{2}}{2} = 2\sqrt{2} \]

Step 6: Confirm against the options.
This diagonal-based route lands on exactly the same value as the direct side-angle area formula, confirming the area is $2\sqrt{2}$ sq cm and ruling out the other three options.

Final Answer:
\[ \boxed{2\sqrt{2} \text{ sq cm}} \]