If the side of a square is `(1)/(2)(x+1)` units and its diagonal is `(3-x)/(sqrt(2))` units, then the length of the side of the square would be

Options

A. `(4)/(3)` units
B. `(1)/(2)` unit
C. 1 unit
D. 2 units

Answer

To solve the problem step by step, we start by using the information given about the square's side and diagonal.

### Step 1: Set up the equations
We are given:
- The side of the square \( s = \frac{1}{2}(x + 1) \) units.
- The diagonal of the square \( d = \frac{3 - x}{\sqrt{2}} \) units.

### Step 2: Relate the side and diagonal of the square
For a square, the relationship between the side length \( s \) and the diagonal \( d \) is given by the formula:
\[ d = s \sqrt{2} \]
Substituting the expressions for \( s \) and \( d \):
\[ \frac{3 - x}{\sqrt{2}} = \frac{1}{2}(x + 1) \sqrt{2} \]

### Step 3: Eliminate the square root
To eliminate the square root, multiply both sides by \( \sqrt{2} \):
\[ 3 - x = \frac{1}{2}(x + 1) \cdot 2 \]
This simplifies to:
\[ 3 - x = x + 1 \]

### Step 4: Solve for \( x \)
Rearranging the equation:
\[ 3 - 1 = x + x \]
\[ 2 = 2x \]
Dividing both sides by 2:
\[ x = 1 \]

### Step 5: Find the length of the side of the square
Now that we have \( x \), we can find the side length \( s \):
\[ s = \frac{1}{2}(x + 1) = \frac{1}{2}(1 + 1) = \frac{1}{2} \cdot 2 = 1 \text{ unit} \]

### Final Answer
The length of the side of the square is \( 1 \) unit.

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