[size=150]What do we mean by the distance between a point and a line segment?[br][br]In the diagram below, what is the distance between point R and line segment AB?[br]Move the sliders to try to make sense of the concept of distance between a point and a line.[br][br]If we let P be a point on the line segment AB, we see that PR is the "distance between R and AB.[br]But there are many locations for P and different distances for P![/size]
[size=150]Move point P along AB.[br][br]Is there a particular position of P on AB where the distance PR is the shortest?[br]What is the angle between PR and AB when the distance is the shortest?[/size]
[size=150]Yes. PR is the shortest when PR is perpendicular to AB (at 90 degrees)[/size]
What do we mean when we want to find the distance between R and AB?[br]How do we obtain this distance?
We are looking for the shortest distance between R and AB.[br]We can get the shortest distance by drawing a line perpendicular to AB which passess through R.[br]The perpendicular distance of R from AB, ie PR is the (shortest) distance between the point and the line!
Explore point Q on line segment BC.[br][br]Adjust the position of Q until you get the shortest distance RQ. (Distance between R and BC)[br]Is RQ = PR?[br][br]Try moving point R and again find the (shortest) distance between R and AB and R and BC[br][br]What can we conclude about the line BX ?
Yes. RQ = PR[br][br]Line BX is in fact the angle bisector of angle ABC.[br]Every point on the angle bisector is equidistant from the line segments AB and BC!