We want to know the shortest distance from the point P to the line (see figure 1). As far as I know it is given by the length of the segment perpendicular to the line that joins the line to the point. Can you check this argument I give is correct? Part A. First let us draw in the segment from the point P to the line that meets the line at 90 degrees (makes a right angle). We call this the perpendicular segment. We call the point where the perpendicular segment meets the line Q (see figure 2). Part B. IMPORTANT!: We prove that the perpendicular segment represents the shortest distance from the point to the line by demonstrating that ANY OTHER SEGMENT from the point P to the line is longer! Part C. To that end consider any point other than Q on the line, call it R. (see figure 3) Part D. We draw in the segment from the point P to the point R. We notice that the points P,Q, and R are the corners of a right angled triangle where the segment from P to R is the hypotenuse and the perpendicular segment (from P to Q) is one of the other sides (see figure 4). Part E. It is well known that the hypotenuse of a right angled triangle is the longest side. Thus we have proved that ANY OTHER SEGMENT is longer than the perpendicular segment. Proof complete.