Question

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 to the longer side?

• A. 1:√3
• B. √3:4
• C. 2:5
• D. 3:4
• E. √3:2

Hint:

In rectangle problems, use the Pythagorean theorem for the diagonal and set up the equation based on the given condition.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Let the shorter side be $a$ and the longer side be $b$. The diagonal is $\sqrt{a^2 + b^2}$. The adjacent sides sum to $a + b$.
Step 2: Key Formula or Approach:
Based on the problem statement: $(a + b) - \sqrt{a^2 + b^2} = \frac{1}{2}b$.
Step 3: Detailed Explanation:
Rearrange the equation:
$a + b - \frac{b}{2} = \sqrt{a^2 + b^2}$.
$a + \frac{b}{2} = \sqrt{a^2 + b^2}$.
Square both sides:
$(a + \frac{b}{2})^2 = a^2 + b^2$.
$a^2 + ab + \frac{b^2}{4} = a^2 + b^2$.
$ab = b^2 - \frac{b^2}{4}$.
$ab = \frac{3b^2}{4}$.
Divide by $b$ (since $b \neq 0$):
$a = \frac{3}{4}b$.
Therefore, $\frac{a}{b} = \frac{3}{4}$.
Step 4: Final Answer:
The ratio of the shorter to the longer side is $3 : 4$.