Why must the compass opening be greater than 1/2 of the segment when construction a perpedicular bisector
so the arcs will overlap, giving two points on the bisector line.
What is the first step in constructing congruent angles
When constructing a perpendicular bisector using a compass, it is important for the compass opening (distance between the two legs of the compass) to be greater than half of the segment being bisected. Here's why:
1. Let's assume we have a line segment AB that we want to bisect perpendicularly. We want to find the midpoint M of the segment AB and draw a line passing through M, perpendicular to AB.
2. To bisect the segment AB accurately, we need to draw arcs on either side of the segment with the same radius. These arcs will intersect at two points, which we'll label as C and D. These points will form a line perpendicular to AB.
3. Now, if the compass opening is less than half of the segment AB, it becomes impossible to draw arcs with the same radius that intersect at points C and D.
4. Imagine if the compass opening is exactly half the length of segment AB. In this case, if we try to draw arcs from points A and B, the arcs will not intersect at two distinct points C and D. Instead, they will either barely touch at a single point, or not touch at all.
5. To ensure that the arcs intersect at two distinct points, and thus allow us to accurately draw the perpendicular bisector, the compass opening needs to be larger than half of the segment AB.
By having a greater compass opening, we have enough "room" to properly draw the arcs and find the points of intersection, allowing us to construct a perpendicular bisector accurately.