In a rectangle, the difference between the sum of the adjacent sides and the diagonal is half the length of the longer side. What is the ratio of the shorter side to the longer side?

Options

A. `sqrt(3):sqrt(2)`
B. `1:sqrt(3)`
C. `2:5`
D. `3:4`

Answer

To solve the problem, we need to establish the relationship between the sides of the rectangle and the diagonal based on the given condition. Let's denote the shorter side of the rectangle as \( b \) and the longer side as \( l \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are given that the difference between the sum of the adjacent sides (which are \( l \) and \( b \)) and the diagonal is half the length of the longer side \( l \).

2. **Set Up the Equation**:
The sum of the adjacent sides is \( l + b \). The diagonal \( d \) of the rectangle can be calculated using the Pythagorean theorem:
\[ d = \sqrt{l^2 + b^2} \]
According to the problem, we can set up the equation:
\[ (l + b) - \sqrt{l^2 + b^2} = \frac{l}{2} \]

3. **Rearranging the Equation**:
Rearranging the equation gives:
\[ l + b - \frac{l}{2} = \sqrt{l^2 + b^2} \]
Simplifying the left-hand side:
\[ \frac{l}{2} + b = \sqrt{l^2 + b^2} \]

4. **Squaring Both Sides**:
To eliminate the square root, we square both sides:
\[ \left(\frac{l}{2} + b\right)^2 = l^2 + b^2 \]
Expanding the left-hand side:
\[ \frac{l^2}{4} + lb + b^2 = l^2 + b^2 \]

5. **Simplifying the Equation**:
Now, we can cancel \( b^2 \) from both sides:
\[ \frac{l^2}{4} + lb = l^2 \]
Rearranging gives:
\[ lb = l^2 - \frac{l^2}{4} \]
This simplifies to:
\[ lb = \frac{3l^2}{4} \]

6. **Finding the Ratio**:
Dividing both sides by \( l \) (assuming \( l \neq 0 \)):
\[ b = \frac{3l}{4} \]
Now, we can find the ratio of the shorter side \( b \) to the longer side \( l \):
\[ \frac{b}{l} = \frac{3l/4}{l} = \frac{3}{4} \]

### Final Answer:
The ratio of the shorter side to the longer side is \( \frac{3}{4} \).