Shortest distance between a point and a line

Grade 10s experienced some productive struggle today as we attempted to determine the shortest distance between a point (4,5) and a line (y=2/5x-6).  Many could guess approximately where the shortest distance would be…. and a few lightbulb moments happened when we realized it would be perpendicular to the line….and we know perpendicular lines have slopes that are negative reciprocals (inverse négative)…so we made an equation for the perpendicular line….and then we needed to solve for the intersection of the original line, and the new line.  We substituted, and solved….and then….and only then….could we use the distance formula.

It was not easy to wrestle with this at first.  Most groups got half way there, and then we looked at the process as a class to be sure we are all on the same page.