Question

7 from unit 1, lesson 3 here are two points in the plane. explain how to construct a line segment that is half the length of segment ab.

Explanation:

Step1: Construct the perpendicular bisector of segment AB

1. Place the compass at point A and draw an arc with a radius greater than half of the length of AB.
2. Without changing the compass width, place the compass at point B and draw another arc. The two arcs intersect at two points.
3. Draw a line through these two intersection points. This line is the perpendicular bisector of segment AB and it intersects AB at its mid - point M. So, \(AM = MB=\frac{1}{2}AB\).

Step2: Construct a segment equal to AM (or MB)

1. Use a straightedge to draw a ray.
2. Place the compass at point A (or B) and open it to the length of AM (or MB).
3. Place the compass at the endpoint of the ray and draw an arc that intersects the ray. The segment from the endpoint of the ray to the intersection point of the arc and the ray is a segment that is half the length of segment AB.

Answer:

First, construct the perpendicular bisector of segment \(AB\) to find its mid - point \(M\) (so \(AM = MB=\frac{1}{2}AB\)). Then, using a compass, transfer the length of \(AM\) (or \(MB\)) to a new ray to get a segment of half the length of \(AB\).