Generalization: Points , (the imageof ), and the originare collinearfor any point in a dilation.
You can provethe generalizationin the LessonExercises.
How do we knowthat a dilationis a similaritytransformation?We wouldhaveto establishthat lengthsof
segmentsare proportionaland that anglesare congruent.Let’s attacktheserequirementsthroughthe distance
formulaand slopes.
Let , , and be pointsin a coordinategrid. Let a dilationhavecenterat the origin
and scalefactor.
Part 1: ProportionalSide Lengths
Let’s look at the lengthsof two segments, , and.
Accordingto the distanceformula,
and