How is constructing a perpendicular bisector similar to constructing an angle bisector? How is it different
Constructing a perpendicular bisector and constructing an angle bisector are similar in that they both involve dividing a line segment or an angle into two equal parts. However, they differ in terms of what they divide and how they achieve the division.
1. Similarity: Dividing into two equal parts
Both constructions aim to divide a geometric figure into two equal parts:
- Perpendicular Bisector: It divides a line segment into two equal parts, creating two segments that are equal in length. The perpendicular bisector also ensures that these two segments are perpendicular to the line segment.
- Angle Bisector: It divides an angle into two equal parts, creating two angles that have the same measurement. The angle bisector also ensures that the two angles are adjacent and share a common vertex.
2. Difference: Object being divided and the construction process
The main difference lies in what is being divided and the process used to achieve the division:
- Perpendicular Bisector: It works by finding the midpoint of the line segment and constructing a perpendicular line passing through that midpoint. To construct a perpendicular bisector:
1. Locate the midpoint of the line segment using a compass or by measuring and marking equal distances from each endpoint.
2. With the compass opened to a length greater than half the line segment, place the compass tip on each endpoint and draw two arcs that intersect above and below the line segment.
3. Draw a straight line connecting the intersection points of the arcs. This line will be perpendicular to the original line segment and divide it into two equal parts.
- Angle Bisector: It works by finding the angle's vertex and bisecting the angle with a straight line. To construct an angle bisector:
1. Place the compass at the vertex of the angle and draw an arc that intersects both sides of the angle.
2. Without adjusting the compass, place the compass tip at each intersection point on the sides of the angle and draw two arcs.
3. Draw a straight line through the vertex and the intersection point of the arcs. This line will divide the angle into two equal parts.
In summary, both the perpendicular bisector and angle bisector constructions involve dividing a geometric figure into two equal parts. The difference lies in what is being divided (a line segment versus an angle) and the process used to achieve the division.