5.2. Perpendicular Bisectors http://www.ck12.org
TABLE5.1:
Statement Reason
1.AC∼=CB Given- 4 ACBis an isosceles triangle Definition of an isosceles triangle
3.^6 A∼=^6 B Isosceles Triangle Theorem - Draw pointD, such thatDis the midpoint ofAB. Every line segment has exactly one midpoint
5.AD∼=DB Definition of a midpoint - 4 ACD∼= 4 BCD SAS
7.^6 CDA∼=^6 CDB CPCTC
8.m^6 CDA=m^6 CDB= 90 ◦ Congruent Supplements Theorem
←→
CD⊥AB Definition of perpendicular lines
10.←→
CDis the perpendicular bisector ofAB Definition of perpendicular bisectorTwo lines intersect at a point. If more than two lines intersect at the same point, it is called apoint of concurrency.
Investigation: Constructing the Perpendicular Bisectors of the Sides of a Triangle
Tools Needed: paper, pencil, compass, ruler
- Draw a scalene triangle.
- Construct the perpendicular bisector for all three sides.
The three perpendicular bisectors all intersect at the same point, called the circumcenter.
Circumcenter:The point of concurrency for the perpendicular bisectors of the sides of a triangle.
- Erase the arc marks to leave only the perpendicular bisectors. Put the pointer of your compass on the circumcenter.
Open the compass so that the pencil is on one of the vertices. Draw a circle. What happens?