In the diagram, the two circles have a common centre O. If the area of the larger circle is 100\(\pi\) and that of the smaller circle is 49\(\pi\), find x

In the diagram, the two circles have a common centre O. If the area of the larger circle is 100\(\pi\) and that of the smaller circle is 49\(\pi\), find x

Answer Details

Let the radius of the smaller circle be r, and the radius of the larger circle be R. The area of a circle is given by the formula A = \(\pi r^2\). Therefore, the radius of the smaller circle is \(\sqrt{\frac{49\pi}{\pi}} = 7\). Similarly, the radius of the larger circle is \(\sqrt{\frac{100\pi}{\pi}} = 10\). Since O is the centre of both circles, the distance between O and the point where the two circles intersect is R - r = 10 - 7 = 3. From the diagram, we can see that x is also equal to this distance, so x = 3. Therefore, the answer is 3.