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To solve the problem, let's break it down step by step.

1. The equation of the curve is \( x^2 + y^2 = 16 \) with \( y > 0 \). This represents the upper semicircle of radius 4 centered at the origin.

2. From any point \( P \) on the semicircle, we draw a vertical line segment \( PP' \) perpendicular to the x-axis. Since \( P' \) is vertically aligned with \( P \) and lies on the x-axis, the coordinates of \( P' \) would be \((x_P, 0)\), where \( x_P \) is the x-coordinate of point \( P \).

3. The midpoint \( M \) of \( PP' \) will have coordinates:
   - \( x_M = x_P \)
   - \( y_M = \frac{y_P + 0}{2} = \frac{y_P}{2} \)

Since \( P \) lies on the curve \( x^2 + y^2 = 16 \), let's denote the coordinates of \( P \) as \( (x_P, y_P) \). Then,
\[ y_P = \sqrt{16 - x_P^2} \]

The midpoint \( M \) will thus be:
\[ (x_P, \frac{\sqrt{16 - x_P^2}}{2}) \]

4. We need to find the equation that describes the locus of \( M \). Let's denote \( x = x_P \) and \( y = \frac{\sqrt{16 - x^2}}{2} \). Hence:
\[ 2y = \sqrt{16 - x^2} \]
\[ 4y^2 = 16 - x^2 \]
\[ x^2 + 4y^2 = 16 \]

5. The equation \( x^2 + 4y^2 = 16 \) represents an ellipse centered at the origin with semi-major axis \( a = 4 \) and semi-minor axis \( b = 2 \).

Comparing this with the given options, the correct choice is:
\[ \boxed{A: \frac{x^2}{16} + \frac{y^2}{4} = 1 \text{ (} y > 0\text{)}} \]

Do you want any further details or have any other questions?

Here are 5 questions you might ask:
1. How do you derive the standard form of an ellipse from its general equation?
2. Can you explain how to find the midpoint of a line segment given its endpoints?
3. How do vertical and horizontal shifts affect the equation of a curve?
4. Can you demonstrate solving a similar problem with different constraints?
5. How would the equation change if the curve was a different shape, like a parabola?

Tip: Remember that converting the coordinates of a point on a curve into the coordinates of a point representing its midpoint involves understanding the geometric properties and algebraic manipulations of the shapes involved.

Explore how to find the locus of midpoints on a semicircle with a detailed solution. This problem involves coordinate geometry, specifically focusing on the properties of ellipses derived from a geometric setup involving a semicircle. Suitable for advanced high school students.

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