Problem Analysis:
The image provides a definition and examples of the
perpendicular bisector of a segment in the context of triangles. The task is to understand the concept of a perpendicular bisector and apply it to the given examples.
#### Definition Recap:
A
perpendicular bisector of a segment is a line (or ray or segment) that is:
1. Perpendicular to the segment.
2. Passes through the midpoint of the segment.
Key points to note:
- The perpendicular bisector does not necessarily start from a vertex of the triangle.
- It can be applied to any segment within a triangle, not just sides.
#### Examples Provided:
1.
Scalene Triangle \( \Delta CDE \):
- Segment \( \overline{CD} \) is shown.
- Line \( \overline{AB} \) is the perpendicular bisector of \( \overline{CD} \).
- \( \overline{AB} \) is perpendicular to \( \overline{CD} \) and passes through its midpoint.
2.
Right Triangle \( \Delta MLN \):
- Segment \( \overline{MN} \) is shown.
- Line \( \overline{AB} \) is the perpendicular bisector of \( \overline{MN} \).
- \( \overline{AB} \) is perpendicular to \( \overline{MN} \) and passes through its midpoint.
3.
Isosceles Triangle \( \Delta POQ \):
- Segment \( \overline{OQ} \) is shown.
- Line \( \overline{PR} \) is the perpendicular bisector of \( \overline{OQ} \).
- \( \overline{PR} \) is perpendicular to \( \overline{OQ} \) and passes through its midpoint.
Solution Explanation:
#### Example 1: Scalene Triangle \( \Delta CDE \)
-
Segment: \( \overline{CD} \)
-
Perpendicular Bisector: \( \overline{AB} \)
-
Verification:
- \( \overline{AB} \) is perpendicular to \( \overline{CD} \) (indicated by the right angle symbol).
- \( \overline{AB} \) passes through the midpoint of \( \overline{CD} \) (point \( B \) is the midpoint).
#### Example 2: Right Triangle \( \Delta MLN \)
-
Segment: \( \overline{MN} \)
-
Perpendicular Bisector: \( \overline{AB} \)
-
Verification:
- \( \overline{AB} \) is perpendicular to \( \overline{MN} \) (indicated by the right angle symbol).
- \( \overline{AB} \) passes through the midpoint of \( \overline{MN} \) (point \( A \) is the midpoint).
#### Example 3: Isosceles Triangle \( \Delta POQ \)
-
Segment: \( \overline{OQ} \)
-
Perpendicular Bisector: \( \overline{PR} \)
-
Verification:
- \( \overline{PR} \) is perpendicular to \( \overline{OQ} \) (indicated by the right angle symbol).
- \( \overline{PR} \) passes through the midpoint of \( \overline{OQ} \) (point \( R \) is the midpoint).
General Observations:
1. In all examples, the perpendicular bisector is correctly identified as a line that is perpendicular to the segment and passes through its midpoint.
2. The perpendicular bisector does not have to originate from a vertex of the triangle, as seen in all three examples.
3. The property of being a perpendicular bisector is independent of the type of triangle (scalene, right, or isosceles).
Final Answer:
The examples correctly illustrate the concept of a perpendicular bisector. The solutions are verified by checking that each line is perpendicular to the segment and passes through its midpoint.
\[
\boxed{\text{The examples demonstrate the correct application of the perpendicular bisector definition.}}
\]
Parent Tip: Review the logic above to help your child master the concept of perpendicular bisector and median.
