answer this question---**Question Stem:**
Five identical semi-circles are arranged as shown.
Find the diameter of one circle
**Diagram Description:**
Type: Geometric arrangement with dimensions.
Description: The image shows five identical semi-circles arranged along a horizontal line. Three semi-circles are placed above the horizontal line, side-by-side, with their diameters resting on the line. Two semi-circles are placed below the horizontal line, positioned in the gaps between the upper semi-circles, also with their diameters resting on the line.
Dimensions labeled are:
- A horizontal arrow below the leftmost upper semi-circle spans its diameter and is labeled "22".
- A horizontal arrow between the first and second upper semi-circles indicates a gap and is labeled "12".
- A horizontal arrow between the second and third upper semi-circles indicates a gap and is labeled "12".
- A horizontal arrow below the rightmost upper semi-circle spans its diameter and is labeled "22".
- A horizontal arrow between the two lower semi-circles indicates a gap and is labeled "16".
**Other Relevant Text:**
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We have five identical semi-circles arranged along a horizontal line. Three semi-circles are above the line, and two are below, fitting in the gaps. We need to find the diameter of one circle using the given measurements.
Let's define our variable. Let d represent the diameter of one identical semi-circle. We can see the upper arrangement has three semi-circles in blue, and the lower arrangement has two semi-circles in red. All semi-circles have the same diameter d.
Let's analyze the upper arrangement. The total length is d plus 12 plus d plus 12 plus d, which equals 3d plus 24. The space between the right edge of the first semi-circle and the left edge of the third semi-circle is d plus 24.
Now let's analyze the lower arrangement. Its total length is d plus 16 plus d, which equals 2d plus 16. The key insight is that the lower arrangement fits exactly in the space between the first and third upper semi-circles. Therefore, we can set up the equation: 2d plus 16 equals d plus 24.
Now let's solve the equation. We have 2d plus 16 equals d plus 24. Subtracting d from both sides gives us 2d minus d equals 24 minus 16, which simplifies to d equals 8. Therefore, the diameter of one semi-circle is 8.