Tools of Geometry Toolkit 1.5-1.6

Today’s Goals: 1. To investigate perpendicular lines &

bisectors.2. To use the midpoint and distance

formulas.

Perpendicular Lines Two lines that intersect to form right

angles.

Symbol: means “is perpendicular to”

Bisectors

Perpendicular bisector (of a segment) A line, segment, or ray that is perpendicular

to a segment at its midpoint Bisects segment into two congruent

segments.

Bisectors

Angle bisector A ray that divides an angle into two

congruent coplanar angles Endpoint is at the angle’s vertex.

Ex.1: Finding Angle Measures

KN bisects JKL mJKN = 5x – 25 mNKL = 3x + 5. Solve for x and find mJKN.

Why do we need the distance formula?

The Distance Formula

212

212 yyxxd

1st coordinate (x1, y1) 2nd coordinate (x2, y2)

Ex.2: Finding Distance Find the distance between T(5,2) and R(- 4,-1). Write answer in simplest radical form.

212

212 yyxxd

Ex.3: Finding the MidpointAB has endpoints A(4,3) and B(8,5)

2,

22121 yyxx

M

A

B

The Midpoint FormulaFind the “average” of each coordinate!!

2,

22121 yyxx

M

1st coordinate (x1, y1) 2nd coordinate (x2, y2)

Ex.4: Finding an EndpointThe midpoint of AB is M(3,4). One endpoint is A(-3,2). Find the coordinates of the other endpoint B.

A

M

2,

22121 yyxx

M

Let the coordinates of B

be (x2, y2)

Geometry Book

P29 2-22E, 29-33

P38 10-12P46 2,16,20,24,26,30

3-D Coordinate System

CHALLENGE PROBLEMS Pg. 48 #60-63

In a three-dimensional coordinate system, the distance between two points (x1, y1, z1) and (x2, y2, z2) can be found using this extension of the Distance Formula.

212

212

212 zzyyxxd

3-D Points#60

A = ________B = ________C = ________D = ________E = ________F = ________G = ________