48. A target board for a dart game has been designed as three concentric circles where each coloured region is the same area. If the radius of the blue circle \( r \mathrm{~cm} \) and the radius of the outer circle is 10 cm , determine the value of \( r \).

To solve this dart game problem, we first need to understand the area of circles. The area of a circle is calculated using the formula \( A = \pi r^2 \). In this case, we have three concentric circles: the inner blue circle, a middle circle, and an outer circle with a radius of 10 cm. Since each coloured region has the same area, we can denote the area of the outer circle, which has a radius of 10 cm, as follows: \[ A_{\text{outer}} = \pi (10)^2 = 100\pi \text{ cm}^2 \] The area of the middle circle (the area between the blue circle and the middle circle) can be represented as: \[ A_{\text{middle}} = A_{\text{outer}} - A_{\text{blue}} = 100\pi - \pi r^2 \] Now, let's denote the area of the blue circle: \[ A_{\text{blue}} = \pi r^2 \] Because the regions are equal, we have: \[ \pi r^2 = 100\pi - \pi r^2 \] This simplifies to: \[ 2\pi r^2 = 100\pi \] Dividing both sides by \( 2\pi \) gives us: \[ r^2 = 50 \] Taking the square root of both sides yields: \[ r = \sqrt{50} = 5\sqrt{2} \text{ cm} \approx 7.07 \text{ cm} \] Therefore, the radius \( r \) of the blue circle is \( 5\sqrt{2} \) cm or approximately 7.07 cm.