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:
Let the sides be \(a\) (shorter) and \(b\) (longer), with \(b > a\). Diagonal \(d = \sqrt{a^2 + b^2}\).

Step 2:
Sum of adjacent sides = \(a + b\). Difference between sum of adjacent sides and diagonal = \((a + b) - \sqrt{a^2 + b^2}\).

Step 3:
Given: \((a + b) - \sqrt{a^2 + b^2} = \frac{b}{2}\).

Step 4:
Rearrange: \(a + b - \frac{b}{2} = \sqrt{a^2 + b^2} \implies a + \frac{b}{2} = \sqrt{a^2 + b^2}\).

Step 5:
Square both sides: \(\left(a + \frac{b}{2}\right)^2 = a^2 + b^2\).

Step 6:
\(a^2 + ab + \frac{b^2}{4} = a^2 + b^2\).

Step 7:
Cancel \(a^2\): \(ab + \frac{b^2}{4} = b^2 \implies ab = \frac{3b^2}{4}\).

Step 8:
Divide by \(b\) (since \(b > 0\)): \(a = \frac{3b}{4}\).

Step 9:
Ratio \(a : b = \frac{3b}{4} : b = 3 : 4\).

Step 10:
Final Answer: 3 : 4