AI Tutor
Imagine you are camping at point AA, need to fill a bucket at a straight river, and then carry it to your campfire at point BB. Where along the river bank should you stop to make your total walking distance as short as possible?
Taking the shortest route from AA to the river and then directly to BB creates a bent path, which is surprisingly tricky to minimize using standard algebra alone.
What if we could straighten out that bent path into a single straight line?

The Reflection Principle

In the 1st century CE, the Greek mathematician Heron of Alexandria showed that light reflecting off a flat mirror always chooses the path of least distance. His insight was simple: reflect one of the target points across the boundary line.
When you reflect point BB across the line LL to create an image point B′B', any point PP on line LL is equidistant to both BB and B′B', meaning PB=PB′PB = PB'.
The total distance AP+PBAP + PB is identical to AP+PB′AP + PB'. By the triangle inequality—which states that a straight line is strictly shorter than any detour—the minimum distance occurs where the segment AB′AB' intersects the boundary line.
How do we turn this geometric reflection into exact coordinate coordinates?

Worked Example: Point on the Axis

Let point A=(1,4)A = (1, 4) and point B=(7,2)B = (7, 2). Find the coordinate of point P(x,0)P(x, 0) on the xx-axis that minimizes the total distance AP+PBAP + PB.
First, reflect B(7,2)B(7, 2) across the line y=0y = 0 to get B′(7,−2)B'(7, -2). The line connecting A(1,4)A(1, 4) and B′(7,−2)B'(7, -2) has slope m=−2−47−1=−66=−1m = \frac{-2 - 4}{7 - 1} = \frac{-6}{6} = -1.