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 is inscribed in a rhombus whose diagonals are \( 12 \) and \( 16 \) units.

Step 1: Understanding the Geometry

• Let the diagonals of the rhombus intersect at point \( O \).
• Since diagonals of a rhombus bisect each other at right angles, half-diagonals are \( 6 \) and \( 8 \).
• The triangle \( \triangle ODC \) is right-angled at \( O \).

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

Step 2: Using Area to Find Radius

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

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

Step 3: Calculate Ratio of Areas

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

 Final Answer:

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

Answer 2

Let the rhombus have diagonals \( d_1 = 12 \) and \( d_2 = 16 \). These diagonals intersect at 90°, so they divide the rhombus into 4 right-angled triangles.

Step 1: Use Pythagoras Theorem

Let the radius of the incircle be \( r \), and let the foot of the perpendicular from the center to side \( DC \) be at a distance \( x \) from one vertex.

Using triangle \( \triangle ODC \): \[ x^2 + r^2 = 6^2 \Rightarrow x^2 + r^2 = 36 \tag{1} \] \[ (10 - x)^2 + r^2 = 8^2 \Rightarrow (10 - x)^2 + r^2 = 64 \tag{2} \]

Step 2: Subtract Equation (1) from (2)

Subtracting (1) from (2): \[ (10 - x)^2 - x^2 = 28 \] Expand: \[ 100 - 20x + x^2 - x^2 = 28 \Rightarrow 100 - 20x = 28 \Rightarrow x = 3.6 \]

Step 3: Find Radius

Substitute \( x = 3.6 \) in (1): \[ r^2 = 36 - (3.6)^2 = 36 - 12.96 = 23.04 \Rightarrow r = \sqrt{23.04} \]

Step 4: Calculate Areas

Area of the circle: \[ \pi r^2 = \pi \times 23.04 \] Area of rhombus: \[ \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 12 \times 16 = 96 \]

Step 5: Compute Ratio

Ratio of the area of the circle to the area of the rhombus: \[ \frac{\pi \times 23.04}{96} = \frac{6\pi}{25} \]

Final Answer:

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