Question

A circle is inscribed in a rhombus with diagonals 12 cm and 16 cm. The ratio of the area of circle to the area of rhombus is

• A. \(\frac{5π}{18}\)
• B. \(\frac{6π}{25}\)
• C. \(\frac{3π}{25}\)
• D. \(\frac{2π}{15}\)

Answer 1

The Correct Option is B

Given: A circle inscribed in a rhombus with diagonals measuring \( 12 \) and \( 16 \) units.

Step 1: Geometric Analysis

• Let the rhombus diagonals intersect at point \( O \).
• The diagonals of a rhombus bisect each other perpendicularly. Therefore, the lengths of the half-diagonals are \( 6 \) and \( 8 \).
• Consider the right-angled triangle \( \triangle ODC \) at vertex \( O \).

Applying the Pythagorean theorem to determine the rhombus side length: \[ DC = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]

Step 2: Radius Calculation via Area

Area of triangle \( \triangle ODC \) (using base and height): \[ \text{Area} = \frac{1}{2} \times 6 \times 8 = 24 \]

Area of the same triangle, expressed using the inradius (\( OE \)) and side \( DC = 10 \): \[ \frac{1}{2} \times 10 \times OE = 24 \Rightarrow OE = \frac{48}{10} = 4.8 \] Thus, the radius of the incircle is \( r = OE = 4.8 \)

Step 3: Area Ratio Computation

Area of the incircle: \[ \pi r^2 = \pi (4.8)^2 = \pi \times 23.04 \] Area of the rhombus: \[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 12 \times 16 = 96 \] Ratio of areas: \[ \text{Required Ratio} = \frac{\pi \times 23.04}{96} = \frac{6\pi}{25} \]

 Final Answer:

(B) \( \boxed{\frac{6\pi}{25}} \)