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
- 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.
- 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.