When constructing a perpendicular bisector why must the compass opening be greater than 1/2 the length of the segment
so the arcs drawn from the ends of the line segment will overlap.
When constructing a perpendicular bisector, the compass opening should be greater than 1/2 the length of the segment because it ensures that the arcs intersect each other in two points. These points of intersection are then used to construct the perpendicular bisector line.
To understand why the compass opening needs to be greater than 1/2 the length of the segment, let's break down the construction process:
1. Start by drawing the given line segment.
2. Place the sharp end of the compass on one end of the line segment and adjust the compass opening to slightly more than half its length.
3. Swing an arc with the compass centered at one end of the line segment.
4. Without changing the compass opening, move the sharp end of the compass to the other end of the line segment and repeat step 3.
5. Where the two arcs intersect, draw a straight line that passes through those two points of intersection. This line is the perpendicular bisector of the original line segment.
By ensuring that the compass opening is greater than half the length of the segment, the arcs will intersect in two distinct points. If the compass opening is exactly half the length of the segment, the arcs will only touch at one point, and it will not be possible to accurately determine the perpendicular bisector.
So, to construct a perpendicular bisector accurately, it is crucial to use a compass opening that is greater than half the length of the segment.
When constructing a perpendicular bisector, the compass opening must be greater than 1/2 the length of the segment to ensure accuracy and precision in creating an accurate bisector.
Here is the step-by-step explanation:
1. Start by placing the compass point on one end of the line segment you want to bisect.
2. Adjust the compass opening to a length greater than half the length of the segment. Imagine that you are drawing two arcs on either side of the segment.
3. While keeping the compass opening the same, swing an arc above and below the line by moving the compass in an arc shape. The arcs should intersect the line on both sides.
4. Without changing the compass opening, place the compass point on the other end of the line segment and repeat the process of swinging arcs above and below the line.
5. The point where the arcs intersect on the line itself is the midpoint of the segment, and it also represents the perpendicular bisector.
If the compass opening is exactly half the length of the segment, it will only intersect the segment in one point, making it difficult to accurately locate the midpoint. By using an opening greater than half the length of the segment, you ensure that the arcs intersect the line on both sides, providing a clear and precise midpoint for the perpendicular bisector.