Minimum length (without calculus)

Algebra Level 2

A B C D ABCD is a rectangle of dimension 5 × 21 5 \times 21 . The points P P and R R (respectively Q Q ) are on the side A D AD (respectively B C BC ).

Find the minimum value of the length B P + P Q + Q R + R C BP+PQ+QR+RC .


The answer is 29.

Chan Lye Lee by Chan Lye Lee , 44, Singapore

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1 solution

Chan Lye Lee
Jan 14, 2020

Make 3 copies of the rectangle. Reflect the red path, as shown. B P + P Q + Q R + R C BP+PQ+QR+RC has the same length as the zig-zag segment in the middle, which the minimum occurs if B C 3 BC_3 is a straight line. Hence the minimum value is ( 4 × 5 ) 2 + 2 1 2 = 2 0 2 + 2 1 2 = 29 \sqrt{(4\times 5)^2+21^2}=\sqrt{20^2+21^2}=29 .

Watch this video for this explanation and another method on vector inequalities.

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