Problem:
In △ABC,∠ABC=90∘ and BA=BC=2​. Points P1​,P2​,…,P2024​ lie on hypotenuse AC so that AP1​=P1​P2​=P2​P3​=⋯=P2023​P2024​=P2024​C. What is the length of the vector sum
BP1​​+BP2​​+BP3​​+⋯+BP2024​​?
Answer Choices:
A. 1011
B. 1012
C. 2023
D. 2024
E. 2025
Solution:
For 1≤i≤2024, the vector sum BPi​​+BP2025−i​​ is the vector pointing from the apex of isosceles right triangle △ABC to the reflection of the apex across the hypotenuse, as seen in the figure below.
Its length is 2 times the height of the triangle, namely 2⋅1=2. Each pair contributes a vector of length 2 , all pointing in the same direction, to the total sum. With 1012 such pairs, the length of the resultant vector is 2⋅1012=(D)2024​.
