When constructing a perpendicular bisector, why must the compass opening be greater than 1/2 the length of the segment?
When constructing a perpendicular bisector, the compass opening must be greater than half the length of the segment because it ensures that the two arcs drawn from the endpoints of the segment intersect each other properly.
To construct a perpendicular bisector, you first need to draw two equal arcs which intersect each other. These arcs are drawn from the endpoints of the original segment. The radius of these arcs is set using a compass, and it should be greater than half the length of the segment.
If the compass opening is exactly half the length of the segment, the arcs drawn from the endpoints will not intersect each other. As a result, the perpendicular bisector cannot be properly constructed. By having a compass opening greater than half the length of the segment, the arcs will intersect each other, allowing for the accurate construction of the perpendicular bisector.
When constructing a perpendicular bisector, the compass opening must be greater than one-half the length of the segment because it ensures that the arcs drawn from the endpoints of the segment will intersect. This intersection point is crucial in order to accurately construct a perpendicular bisector.
Here's a step-by-step explanation of why the compass opening should be greater than one-half the length of the segment:
1. Start by placing the compass point on one endpoint of the segment.
2. Open the compass wider than one-half the length of the segment. This ensures that the compass can draw arcs with a radius longer than half the length of the segment.
3. Without changing the compass width, draw two arcs on either side of the segment. These arcs should be big enough to intersect.
4. Without changing the compass width, move the compass point to the other endpoint of the segment.
5. Again, draw two arcs on either side of the segment, intersecting with the previous arcs. The intersection points of the arcs determine the midpoint of the segment.
6. Finally, use a straightedge to connect the midpoints of the segment and draw a line. This line will be the perpendicular bisector, which divides the segment into two equal halves and forms a right angle with it.
If the compass opening is exactly one-half the length of the segment, the arcs drawn from the endpoints may not intersect, resulting in an inaccurate perpendicular bisector construction. It is essential to open the compass wider than one-half the segment length to ensure accurate construction.
When constructing an angle bisector, why must the arcs intersect?
When constructing a perpendicular bisector, the compass opening needs to be greater than half the length of the segment to ensure accuracy and precision in the construction.
Here's how you would construct a perpendicular bisector with a compass and straightedge:
1. Start by drawing the given line segment with endpoints A and B.
2. Place the compass tip at point A and adjust the compass opening to a length greater than half the length of the segment AB. Let's call this length "r".
3. Keeping the compass opening fixed at length r, swing arcs above and below the line segment AB. These arcs should intersect the line segment at two different points. Let's call these points C and D.
4. Without changing the compass opening, place the compass tip at point C and swing an arc above the line segment.
5. Repeat step 4, but this time place the compass tip at point D and swing an arc below the line segment.
6. You should notice that the two arcs intersect at a point above the line segment (let's call this point E) and a point below the line segment (let's call this point F).
7. Using a straightedge, draw a line passing through points E and F. This line is the perpendicular bisector of the line segment AB.
The reason the compass opening needs to be greater than half the length of AB is to ensure that the arcs intersect in two different points, C and D, on AB. If the compass opening was exactly half the length of AB, the arcs would only intersect at point A, resulting in a less accurate construction.
By using an opening greater than half the length, we guarantee that the arcs will intersect at two distinct points, allowing us to accurately construct the perpendicular bisector.