128 Translations; axial symmetries; isometries Ch. 8
Exercises
8.1 IfT is the set of all translations ofΠ, show that(T,◦)is a commutative
group.8.2 IfIis the set of all isometries ofΠ, show that(I,◦)is a group.8.3 Given any half-lines[A,B,[C,Dshow that there is an isometryfwhich maps
[A,Bonto[C,D.8.4 Show that each of the following concepts is an isometric invariant:-interior
region of an angle-support, triangle, dividing a pair of points in a given ratio,
mid-point, centroid, circumcentre, orthocentre, mid-line, incentre, parallelo-
gram, rectangle, square, area of a triangle, circle, tangent to a circle.8.5 For any linel,sl[C(O;k)] =C(sl(O);k)so that, in particular, ifO∈lthen
sl[C(O;k)] =C(O;k).