Reflection Method
Lesson · Intermediate
Geometry: Plane Geometry
A and B are two fixed points. Line l is fixed. Point P lies on l and can move. Find min{PA + PB}.
Approach: Reflect point A about l to get A'.
By Triangle Inequality. Triangle Inequality in triangle A'BP:
Therefore, Q is the point of interest.
Suppose ray AP reflects and goes to Q in square ABCD.
We can reflect the squares and find corresponding intersection points.
Points A and B lie on the same side of line l. Their perpendicular distances from l are 3 and 5, and the distance between their perpendicular projections onto l is 12. Point P lies on l. Find the minimum value of PA + PB.
A path starts at A, touches the x-axis at P, then touches the y-axis at Q, and finally reaches B. Find the minimum possible value of
Reflect A and B across the x-axis and y-axis, respectively.
Equality is attainable when P = P₁ and Q = Q₁.