How many points in a plane are in the locus of all points equidistant from two parallel lines in the plane?
a.0
b.1
c.2
d.infinitely many
D?
Yes. The locus is a line, and there are an infinite number of points in a line.
Thank you!
The correct answer is d. infinitely many.
To understand why, let's first consider the definition of the locus of points equidistant from two parallel lines.
The locus of points equidistant from two parallel lines is a line that lies equidistant from the two lines and is parallel to them. This line forms an infinite number of points.
No matter how close the two parallel lines are, there will always be an infinite number of points that are equidistant from them. Therefore, the locus of all points equidistant from two parallel lines in a plane contains infinitely many points.
To determine how many points are in the locus of all points equidistant from two parallel lines in a plane, we can use the concept of perpendicular bisectors.
When two lines are parallel, any point on the perpendicular bisector of the segment joining them will have the same distance to both lines. Therefore, to find the locus of points equidistant from the two lines, we need to find the perpendicular bisector of the segment joining them.
The perpendicular bisector of a segment is the line that is perpendicular to the segment and passes through its midpoint. In this case, since the two lines are parallel, the segment joining them is also parallel to the two lines. Hence, the midpoint of this segment lies on the line equidistant from the two lines.
The perpendicular bisector of a segment is unique, and it intersects the given segment at its midpoint. So, the locus of points equidistant from two parallel lines in a plane is a unique line.
Therefore, the answer to the question is:
b. 1