We have a rectangle ABCD where AD equals 2 and AB equals 4. AC is the diagonal. Points E and F are moving points on sides AB and CD respectively. Line segment EF is perpendicular to diagonal AC at point M. We need to find the minimum value of AF plus CE.
Let's establish a coordinate system. We place A at the origin, so A is at coordinates zero comma zero. Then B is at four comma zero, C is at four comma two, and D is at zero comma two. The diagonal AC has the equation y equals one half x, so its slope is one half. Since EF is perpendicular to AC, the slope of EF must be negative two.
Now let's find the constraint relationship. We parametrize point E as coordinates x comma zero on side AB, and point F as coordinates x F comma two on side CD. The slope of EF is two divided by x F minus x. Since EF is perpendicular to AC, this slope equals negative two. Solving this equation, we get x F equals x minus one. This gives us the constraint relationship between the positions of E and F.
The key insight is to use auxiliary points to transform the problem. We introduce point H on CD with the same x-coordinate as E, so CE equals BH. We also define A prime at coordinates one comma zero, so AF equals A prime H. Using the reflection principle, we reflect B across line y equals two to get B prime at coordinates four comma four. The minimum value of AF plus CE equals the distance from A prime to B prime, which is five.
To summarize our solution: We established a coordinate system with A at the origin, found the constraint relationship from the perpendicular condition, used auxiliary points for transformation, and applied the reflection principle. The minimum value of AF plus CE is five.