The diameters of two circles are the side of a square and the diagonal of the square. The ratio of the areas of the smaller circle and the larger circle is :

সঠিক উত্তর

$\frac{1}{2}$

সঠিক উত্তর: $\frac{1}{2}$

Question

The diameters of two circles are the side of a square and the diagonal of the square. The ratio of the areas of the smaller circle and the larger circle is :

Accepted Answer

সঠিক উত্তর: $\frac{1}{2}$। ব্যাখ্যা: MCQ Explanation: Ratio of Areas of Two Circles Understanding the Ratio of Areas of Two Circles Question: The diameters of two circles are the side of a square and the diagonal of the square. The ratio of the areas of the smaller circle and the larger circle is: $\frac{1}{2}$ $\frac{1}{4}$ $\frac{\sqrt{2}}{\sqrt{3}}$ $\sqrt{3}$ Given that the correct answer is $\frac{1}{2}$, let's delve into the detailed reasoning behind this: 1. Understand the Problem We have two circles with their diameters equal to the side and the diagonal of a square, respectively. Let's define the side of the square as s. 2. Calculating the Diameters - The diameter of the smaller circle is the side of the square, which is: $d_{small} = s$ - The diameter of the larger circle is the diagonal of the square. Using the Pythagorean theorem, the diagonal d of a square with side s is given by: $d_{large} = \sqrt{s^2 + s^2} = s\sqrt{2}$ 3. Calculating the Radii Next, we find the radii of the circles. The radius is half of the diameter: Radius of the smaller circle: $r_{small} = \frac{s}{2}$ Radius of the larger circle: $r_{large} = \frac{s\sqrt{2}}{2}$ 4. Calculating the Areas The area of a circle is given by $A = \pi r^2$. Therefore: Area of the smaller circle: $A_{small} = \pi \left(\frac{s}{2}ight)^2 = \pi \frac{s^2}{4}$ Area of the larger circle: $A_{large} = \pi \left(\frac{s\sqrt{2}}{2}ight)^2 = \pi \frac{(s^2 \cdot 2)}{4} = \pi \frac{2s^2}{4} = \pi \frac{s^2}{2}$ 5. Finding the Ratio of the Areas To find the ratio of the area of the smaller circle to the larger circle, we divide the area of the smaller circle by the area of the larger circle: $	ext{Ratio} = \frac{A_{small}}{A_{large}} = \frac{\pi \frac{s^2}{4}}{\pi \frac{s^2}{2}} = \frac{\frac{s^2}{4}}{\frac{s^2}{2}} = \frac{1}{2}$ Conclusion Hence, the ratio of the areas of the smaller circle to the larger circle is $\frac{1}{2}$. MathJax.typeset(); এই প্রশ্নটি সহ গণিত বিষয়ের আরও প্রশ্ন মেধাবীর ফ্রি ইন্টারেক্টিভ কুইজে সাথে সাথে ফলাফল ও ব্যাখ্যাসহ অনুশীলন করা যায়।

The diameters of two circles are the side of a square and the diagonal of the square. The ratio of the areas of the smaller circle and the larger circle is :

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এবার দেখা যাক এই প্রশ্নের উত্তরটা দিতে পারেন কিনা।

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