Geometry with Trigonometry

Sec. 1.5 Revision of geometrical concepts 15

A

B

D C

Figure 1.12. A convex quadrilateral.

1.5.3 Distance; degree-measure of an angle

1. With each pair(A,B)of points we associate a non-negative real number|A,B|,
   called thedistancefromAtoBor thelengthof the segment[A,B]. In all cases|B,A|=
   |A,B|. By observation, given any non-negative real numberk, and any half- line[A,B
   there is a unique pointP∈[A,Bsuch that|A,P|=k.

A

k P

Figure 1.13. Laying off a distancek.

P

Q

R

x

y

x+

y

Addition of distances.

By observation, ifQ∈[P,R]then|P,Q|+|Q,R|=|P,R|.In all cases|A,A|=0, while
|A,B|>0ifA=B.

2. Given distinct pointsAandB, choose the pointC∈[A,Bso that|A,C|=^12 |A,B|.
   ThenCis betweenAandBand

|C,B|=|A,B|−|A,C|=|A,B|−

1

2

|A,B|=

1

2

|A,B|=|A,C|.

The pointCwhich is on the lineABand equidistant fromAandB, is called the
mid-pointofAandB. It is also called the mid-point of the segment[A,B].

A

C

B

+

+

Figure 1.14. Mid-point ofAandB.