Geometry 6 1 Perpendiculars and Bisectors What are

Geometry 6. 1 Perpendiculars and Bisectors What are the characteristics of points on a perpendicular bisector?

Goals n n Know what a perpendicular bisector is. Use the Perpendicular Bisector Theorem. Use Angle Bisector Theorem. Measure the distance to a line. 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors 2

6. 1 Perpendiculars and Bisectors 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors 3

The perpendicular bisector of a segment can be a segment. R A B S RS is a bisector of AB. 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors 4

The perpendicular bisector of a segment can be a line. R A B S RS is a bisector of AB. 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors 5

The perpendicular bisector of a segment can be a ray. R A B S RS is a bisector of AB. 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors 6

The perpendicular bisector of a segment can be a plane. K A B K is a bisector of AB. 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors 7

Equidistant Points n A point is equidistant from two points if its distance to each point is the same. R A B S n n R is equidistant from A and B. S is also equidistant from A and B. 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors 8

Theorem 6. 1 (Perpendicular Bisector Theorem) AR BR n If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. R A B S AS BS 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors 9

Theorem 6. 1 Proof n n The two right triangles are congruent by SAS. This means AR = BR by CPCTC. 15 December 2021 R A Geometry 6. 1 Perpendiculars and Bisectors B 10

Theorem 6. 2 (Converse of Perpendicular Bisector Thm. ) n If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. 15 December 2021 R A Geometry 6. 1 Perpendiculars and Bisectors B 11

Example 1 R 14 ? 14 A 15 December 2021 B Geometry 6. 1 Perpendiculars and Bisectors 12

Example 2 R 7. 8 8 A S B Is RS the perpendicular bisector of AB? No, RA RB. 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors 13

Distance from a point to a line. n Defined as the length of the perpendicular segment between the point and the line. 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors 14

Distance from a point to a line. n Defined as the length of the perpendicular segment between the point and the line. This is the distance from the point to the line. 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors 15

Equidistant from two lines. n Point J is equidistant from lines m and n. m J 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors n 16

Theorem 6. 3 (Angle Bisector Theorem) n If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. A D C B 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors 17

Theorem 6. 4 (Converse of Angle Bisector Theorem) n If a point is equidistant from the two sides of the angle, then it is on the bisector of an angle. A D C B 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors 18

Example 3 15 If AD = 15, then DC =____. A D C B 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors 19

Example 4 Is D on the bisector of the angle? Yes 4 D 4 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors 20

Example 5 Is E on the bisector of the angle? NO! 5 E 15 December 2021 5 Geometry 6. 1 Perpendiculars and Bisectors 21

Summary n A point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. AR BR R A B S AS BS 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors 22

Summary n A point on the bisector of an angle is equidistant from the sides of the angle. A D C B 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors 23

Summary n The distance from a point to a line is the measure of the perpendicular segment. 15 December 2021 Geometry 6. 1 Perpendiculars and Bisectors 24