The angle between sides of a rhombus whose v2 times sides is mean of its two diagonal, is equal to:`a)30^@ (b) 45^@ (c) 60^@(d) 90^@`

Options

A. ` 300^(@) `
B. ` 45^(@) `
C. ` 60^(@) `
D. ` 90^(@) `

Answer

To solve the problem, we need to determine the angle between the sides of a rhombus given that \( \sqrt{2} \) times the side is the mean of its two diagonals. Let's break down the solution step by step.

### Step 1: Understand the properties of a rhombus
A rhombus has the following properties:
- The diagonals bisect each other at right angles (90 degrees).
- The diagonals are not necessarily equal.

### Step 2: Set up the problem
Let the lengths of the diagonals be \( x \) and \( y \). The sides of the rhombus can be denoted as \( z \). According to the problem, we have:
\[ \sqrt{2} \cdot z = \frac{x + y}{2} \]
This means that \( z \) is related to the diagonals.

### Step 3: Apply the Pythagorean theorem
Since the diagonals bisect each other at right angles, we can use the Pythagorean theorem in one of the right triangles formed by the diagonals:
\[ z^2 = \left(\frac{x}{2}\right)^2 + \left(\frac{y}{2}\right)^2 \]
This simplifies to:
\[ z^2 = \frac{x^2}{4} + \frac{y^2}{4} = \frac{x^2 + y^2}{4} \]

### Step 4: Substitute \( z \) from the mean equation
From the mean equation, we can express \( z \):
\[ z = \frac{x + y}{2\sqrt{2}} \]
Now, squaring both sides gives:
\[ z^2 = \left(\frac{x + y}{2\sqrt{2}}\right)^2 = \frac{(x + y)^2}{8} \]

### Step 5: Set the two expressions for \( z^2 \) equal
We have two expressions for \( z^2 \):
1. \( z^2 = \frac{x^2 + y^2}{4} \)
2. \( z^2 = \frac{(x + y)^2}{8} \)

Setting them equal gives:
\[ \frac{x^2 + y^2}{4} = \frac{(x + y)^2}{8} \]

### Step 6: Cross-multiply and simplify
Cross-multiplying yields:
\[ 8(x^2 + y^2) = 4(x^2 + 2xy + y^2) \]
Expanding and simplifying gives:
\[ 8x^2 + 8y^2 = 4x^2 + 8xy + 4y^2 \]
\[ 4x^2 + 4y^2 - 8xy = 0 \]
Dividing through by 4 results in:
\[ x^2 + y^2 - 2xy = 0 \]

### Step 7: Factor the equation
This can be factored as:
\[ (x - y)^2 = 0 \]
Thus, we find:
\[ x = y \]

### Step 8: Conclusion about the rhombus
If \( x = y \), then the diagonals of the rhombus are equal, meaning the rhombus is actually a square. In a square, the angle between adjacent sides is \( 90^\circ \).

### Final Answer
The angle between the sides of the rhombus is \( 90^\circ \).