what is the locus of the midpoints of the line segments joining a given external point to all points of a plane?
(Locus is set in space aka 3d)
To understand the locus of the midpoints of the line segments joining a given external point to all points of a plane, let's break it down step by step:
1. Define the problem: We are given a plane and an external point. We want to find the locus (set of all possible positions) of the midpoints of the line segments that connect the external point to all points on the plane.
2. Understand the concept: The midpoint of a line segment is a point that divides the line into two equal halves. In this case, we want to find the set of all midpoints for all line segments connecting the external point to any point on the plane.
3. Analyze the situation: In space (3D), we can imagine the given plane as a flat surface extending infinitely in all directions. The external point can be positioned anywhere outside the plane.
4. Find the locus: The locus of the midpoints of the line segments can be described as a plane parallel to the given plane and passing through the external point. This means that all midpoints lie on this parallel plane.
The reason for this is that the midpoints of the line segments will be equidistant from the given plane and the external point, forming a new plane that is parallel to the original plane.
5. Visualization: Imagine a plane floating in space, and an external point anywhere outside that plane. Now, visualize all the line segments connecting the external point to every point on the plane. The midpoints of these line segments will form a parallel plane passing through the external point.
In conclusion, the locus of the midpoints of the line segments joining a given external point to all points of a plane in 3D space is a plane parallel to the given plane, passing through the external point.